Yui's interruption during the climax of the baseball match If Hinata caught the ball and disappeared then, Yui would most likely be unable to resolve her main regret in life, the inability to be cared for by someone else. Thinking that her disability would prevent anyone from caring for her, she thought she would end up alone without her mother, but Hinata promises to marry her, even in the next life, and even if she was still unable to move in her next life.
No one who commits suicide ends up in purgatory, despite presumably having many regrets. Why? Because their need to move on outweighed their regrets.
I kind of interpreted that scene as Specifically Suspicious Denial, insinuating that Yuri herself committed suicide; Otanashi just asks her how she died and she goes on an unrelated outburst about how no one in the SOS committed suicide.
Watching the anime, you may have noticed that Hinata looks and kind of acts like a standard protagonist. That's because he is the protagonist of the manga, which started after the anime.
Actually, the manga is an adaptation of the light novels, that started before the anime.
The very title of the series. 'Angel' clearly refers to Kanade, but the 'Beats' part, not to mention the heart monitor in the logo, refer to the fact that Kanade got Otonashi's heart as an organ transplant back while she was alive.
"My soul, Your beats".
Also, why does Kanade act in such an awkward manner? Related to Fridge Horror below; she never graduated from school. Even (Japanese) middle school; this means that she has the mind of someone 12 or so years old, tops.
I got the impression her awkward manner was because everybody she ever made friends with vanished. You'd be pretty distant too if all your friends ceased to exist. Why make new ones? You know exactly how that's going to end.
When in episode 9 Otonashi signs his organ donor card as his last act in life, guess whose theme is playing at the time...? Just to state it directly: Kanade, we discover in episode 13, later receives Otonashi's heart as a result of this action, and the entire reason she's in the afterlife is to thank the person who gave her that heart.
And while we're on the subject of Kanade's theme, why does it end with such a sudden, dark chord? The theme mirrors her life. She had a chronic heart problem; most of it was probably weak and ethereal, as most of the melody is. Noticeably, the theme nearly ends at one point in a similar fashion to how it actually does end; this was where her condition degenerated to the point of needing a transplant, at which point she receives Otonashi's heart, allowing a recovery. But sadly, as happens in many such cases in real life, there were complications subsequently, and so young (and we do mean young; see Fridge Horror below) Kanade's life ended rather abruptly: the final, unexpectedly minor chord. Truly a brilliantly composed piece of music.
On the subject of Kanade's illness, her appearance as what some would call is a generic Loli actually has a reason in this series. People who receive donor hearts when they're young don't tend to grow very much physically. But Kanade died at 16 (Or so the official information states), and Otonashi was 17. Due to heart donors and receivers needing to be VERY similar, this would mean that she got the heart, minimum, at about the age of 15. She looks very pre-pubescent, so 12ish I suppose. That would mean her affection to Otonashi is somewhat mal-directed if she'd had multiple heart donations from the time she was 12. So it's more likely she had a weak heart her entire life (Which also fits in with her unsociable personality). Her lack of School activity is potentially due to this illness. On that subject again, Kanade was in this afterlife PRIOR to Otonashi, and she had apparently made many other people disappear... which implies that someone is choosing when and where to place these souls. If they had appeared in order of death, not nearly as many people would have disappeared prior, and Kanade wouldn't seem AS distant.
On the subject of Kanade appearing prior to Otonashi is pretty simple: There's no given time or place in purgatory. For all we know each character could have lived in different millenniums (Even though most did live around the same time).
Otonashi taking his sister out of the hospital so she could see the Christmas festival first-hand may have prevented her from entering Purgatory herself.
In the non-canon OVA, Yuri is so furious about failing that she orders the rest of the Battlefront to go without food and water for an entire week. Everyone except Yuri dies, something she shows no remorse over and even seems to take sadistic pleasure in (she makes the excuse that they'll all come back to life, but that doesn't mean they don't feel agonizing pain as they die). The credits ED shows all the other characters fading until only Yuri is left, followed by the words "Bad End". The characters will just come back to life, so why is it a Bad End? Because the outcome of the story hinges on Yuri's HeelFace Turn thanks to The Power of Friendship she shares with everyone else, and if this route was taken (as if in a visual novel game), she would have crossed the Moral Event Horizon. The Power of Friendship as an outcome is definitely out after Yuri lets her supposed friends die a slow, painful death from starvation and dehydration and feel nothing over it. Basically, this OVA shows what would happen if Yuri went too far beyond the line, with the last line of dialogue spoken (by a dying Yusa) even being about how she is absolutely "the villain".
When Angel goes to attack Guild the SSS members are able to escape by blowing the entire underground up and trapping her under tons of rock. Since Death Is Cheap she must have woken up under it all or survived the fall. Either way she had to dig herself out.
I always figured that if you died in a difficult enough position or didn't wake up in a while, you would respawn in the infirmary. Otonashi did after he was stabbed by Angel. If the SSS had retrieved him, he would have probably ended up in the principal's office instead, like when he got hammered out the window later on. Of course, there's the possibility that Angel carried him there...
After blowing up Guild in episode 2, Angel was covered in boulders and molten rock. Realize that from what we're told, people regenerate and respawn right where they were before, meaning that to get out she needed to clamber through hundreds of tons of rock crushing her and swimming through lava.
The realization that if purgatory really worked this way, Naoi's view on things would probably be a lot more common.
Even though Kanade received Otonashi's heart via transplant, she had to have died during childhood to end up at the school.
"You've never been to a graduation, have you?"
Alternately, it simply means that she was always too ill to attend school.
Angel has been shot enough times by the SSS to take a bullet from Otonashi in the stomach in episode 1 and barely even flinch. Remember, this is a human girl we're talking about, Angel Player enhancements or not.
Kanade wanted to thank Otonashi for receiving his heart in a transplant. This might come off as some odd Fridge Logic, but that would mean she didn't get to enjoy her heart for long.
Since most characters appear young, it could mean that most of the students of the school and the SSS that aren't NPCs could've died around the age they appear to be in. Otonashi and Iwasawa and more of the SSS including Kanade confirms this.
Any person who enters this afterlife faces two possible final fates: you either disappear, or you get turned into an NPC. Even if there is no Shadow invasion going on at the moment, there is always the possibility of one occurring in the future, and you have unlimited time to wait. With potential infinite time to live in that afterlife, and since the probability of becoming an NPC is always nonzero (as the Shadows could always come back), it is therefore a mathematical certainty that you will become an NPC eventually, unless you disappear before this happens. If you disappear, you just reincarnate, live another life, and one day die again, returning to the afterlife. Disappearing therefore only delays the threat of becoming an NPC. There is no true escape from this threat. Being an NPC is not confirmed to be reversible in all cases. One guy is said to have returned from it and successfully disappeared, but he was a recently turned NPC, and we didn't see how this happened. Most of the NPCs shown around the school are not known to be recoverable, but are mostly written off as lost causes. Unless there is more to the afterlife than the setting seen in the series, all of humanity is doomed to inevitably become NPCs.
Although since this afterlife only contains those that died during adolescent age with unfulfilled life and that they can pass through it, the chance of becoming NPCs is very, very low. Not to mention that there is usually a 'guardian' of some sort in it (Kanade at the start of the series and continued by Otonashi in an alternate ending which may or may not take place before the final scene of the final episode if reincarnation is also not bound by time just as it's possible to wander around another period of the afterlife as proved by Kanade). Unless of course, other afterlives contains Shadows too.
Consider the following, a man with no regrets of his life gets stuck in Purgatory where he starts to help others move on from their life, he originally had no memories because he didn't have his own heart. Later he falls in love with the one that has his heart, and watches her move on. Now understand this, a man with no heart and no regrets can't move on to the afterlife. Basically all he did for his friends (and those that will come after) will be but a dream to him. This troper considers it both heartbreaking and horrifying how this turns out in context.
Additionally, we learn in episode 12 that the programmer behind Angel Player came to the purgatory with amnesia due to living his life for the sake of another. Afterwards he fell in love with a girl who disappeared after coming to terms with her past regrets. The programmer then waited for her return in the purgatory for a long time, resulting in him losing his sanity. As a result, he created Angel Player with the patch that if a situation like his occurs, then the Shadows would appear and turn any human into an NPC, including himself. Otonashi's situation is similar where he falls in love with Kanade, who disappears after thanking him for donating his heart to her. If the alternate ending happens where Otonashi remains in the purgatory and is waiting for Kanade, then he might lose his sanity due to waiting for a long time. As a result, he might create the Angel Player program again and turn himself along with everybody else into an NPC, should a person enter the purgatory the same way he did. In other words, history might repeat itself.
So if you donate your heart you will wake up in purgatory with no memories. There's no guarantee of meeting the person who has your heart so no guarantee you'll ever get them again. Man, I'm going, I need to change a form at my doctor's...
Additionally since Otonashi woke up in purgatory without a heart, then he died without heart? Does that mean he was still alive when they harvested his organs? Alternatively if they did a face transplant, would he wake up in purgatory without a face?
The big reveal of Otonashi's backstory and his relationship to Kanade actually creates a massive Plot Hole in the story. According to Otonashi's backstory, after suffering internal injuries in a train crash, he signed off to give hospitals the abilities to use his organs for transplants. His heart was used to extend Kanade's life, which is fair enough. Despite the fact that Otonashi died before Kanade did, she ended up going to the After Life/Limbo before he did. And for an undisclosed amount of time before he did too, since all of the other SSS members were already familiar with Kanade.
From what we see in his backstory, Otonashi didn't have any regrets when he died so he didn't end up in the school/limbo. Conversely, Kanade's regret in life was that she never got to thank the person whose heart was given to her. Combine that with the fact that time means nothing in the afterlife (as evidenced by Shiina apparently having died centuries ago), and this results in Otonashi having been taken from the afterlife and dropped at the school for the purpose of alleviating Kanade's regret.
Further more after the revelation one is left to wonder why Otonashi is stuck in the limbo even after Kanade finally made her own peace? The only reason he was drawn there in the first place was because of Kanade's regerets for never thanking him, as he actually died feeling fulfilled with his own life.