The option to save Chloe and why it exists
... a k a Max is not Commander Shepard
At the end of Day 5, you are presented with a binary choice — Save Arcadia Bay or Save Chloe. The game sends you clue after clue, hint after hint that Chloe is a Doom Magnet and trying to save her only causes bad things to continue to occur. Almost every day has you trying to save Chloe's life, be it Nathan with a gun, Chloe shooting at junk, Chloe playing on the train tracks, Chloe becoming a quadriplegic who is slowly withering away, Chloe being killed by Mr. Jefferson and so on. The game is telling you that YOU DONE FUCKED IT UP!! when you saved Chloe the first time, and the only way to prevent the tornado from wrecking the town is to go back to Day 1 and let Nathan kill Chloe. Now, accepting that realization and fixing the timeline would have in itself been an excellent ending — by providing An Aesop about causality, how linear time strongly enforces it, and the disastrous pitfalls of messing with it. So why does the game allow Max to not learn that valuable lesson, instead opting to save Chloe?
It is because Max would have to learn the extremely Family-Unfriendly Aesop that sometimes an innocent life must be sacrificed for the greater good.
Now, consider who is this particular Aesop useful to? It is usually people with tremendous responsibility and a great deal of authority over people, who will assist that leader. People like ship's captains, mission commanders, field commanders, secret agents, law enforcement superagents, ICU personnel doing triage, emergency medicine specialists, governors, presidents, prime ministers etc. For the most part, these people volunteered for the job, or were otherwise qualified for it, thus they are receptive to this lesson. Now consider what Max Caulfield is? She is a shy introverted photography nerd who is in high school. She is just concerned with navigating high school cliques, not navigating a spaceship through a mine field. She just wants to make it through her Science, Lit and Art classes, not make it through a Zerg Rush, a Macross Missile Massacre and a Beam Spam. The most risky thing she would have ever considered doing, is to dip in the pool with Chloe and maybe steal a kiss, not say, infiltrating an enemy base and stealing vital intelligence. She is content to trade barbs with Victoria, not trading broadsides across the bow. Max is even an Actual Pacifist who Does Not Like Guns. She realizes, like other teen superheroes such as Spider-Man do, that with her ability Comes Great Responsibility. But should she be expected to learn the harsh truth about the brutal calculus? Should she be expected to at that age bear the burden of sacrificing Chloe, knowing that doing so will save the town? Or would it be ok for Max the high schooler to just want to be with Chloe?
Ultimately that is the choice the game puts to the player. Should you as the player act like you are playing Deus Ex or Mass Effect 3 and make the cold calculating choice? Or should you realize that you are role playing as an eighteen year old introverted art student and act with the heart?
THAT is the choice in the end.
The ending cutscenes play out accordingly. Sacrifice Chloe and all you see Max do is cry — she weeps uncontrollably, huddled in the corner as Chloe gets shot, then is shown crying at the funeral. This cutscene is a What the Hell, Player? montage saying, "Look! Because you made her act according to the brutal calculus when she wasn't prepared to handle it, she is now broken!" The other cutscene where she saves Chloe shows a happier Max. Yes, the town was destroyed, but in the end she didn't lose her true companion. She isn't as broken about Arcadia Bay being destroyed and whatever loss of life may have occurred because they are abstract things, while Chloe is a concrete real thing that she didn't have to sacrifice.