These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
The 456: really Extra Sufficiently Advanced Aliens, or just bluffing? They clearly do have advanced virology, but we never see any other technology, and the only physical defense they rely on was mundane bulletproof glass. Making all the children of Earth speak in unison isn't much more impressive than what the Sycorax did, and they were bluffing. This ambiguity only makes the story even more tragic, but left some viewers wondering if UNIT or the British government had really tried everything before surrendering.
Or even worse, what if the 456 Ambassador was the only individual responsible for the entire incident? A junkie who'd conned the entire Human race into supplying him with children under threat of the rest of his species, who might have no idea what he was up to?! Much like the Slitheen are to the Raxacoricofallapatorians, this could be simply one bad egg in an otherwise peaceful species?!
Broken Base: Ianto's death (and/or the way it was handled) and whether Jack's actions were justifiable in context.
The 456, while being Starfish Aliens, are very definitely examples of this trope. After taking control of all the world's children in order to communicate, it turns out that they use human children in some nasty symbiotic way in order to get high and are bargaining to take 10% of the world's children to use as drugs or else they Kill All Humans. This is a protection racket, and they would almost certainly have been back for more later. One child is seen hooked up to one of them and it's shown that he's been a human reefer for over 40 years. It's really twisted and nasty.
Prime Minister Brian Green calmly allows 10% of the world's children to be sold as narcotics. He orders Frobisher, the man who's been most loyal to him, to give up his own children just to make the cover story he's created realistic. This leads him to shoot his family and himself to spare them the horror. After all the horror and pain, Green's first thought is how he can blame the Americans.
Die for Our Ship: Gwen is perceived as getting in the way of Jack/Ianto, especially after Ianto's death.
Growing the Beard: Very contentious, but this miniseries was the first real critical and ratings success for the show.
Hilarious in Hindsight: Gwen's "Where is the Doctor" angst, in light of Peter Capaldi's casting as the Twelfth Doctor.
Prime Minister Green, when he orders Frobisher, his most loyal ally, to sacrifice his own children to the 456.
"I'm sorry John. I'm really very sorry...and I'm really very busy."
The ministers of the government choosing to protect their own children no matter what and then taken even further when they all agree to sacrifice the poor and disadvantaged children to save the respectable middle class ones. Though they still don't end being as unsympathetic as Green, since they at least admit to their bias and aren't exactly happy about their actions.
Nope. The cheap shots begin religious and non-religious factions tend to go both ways in media. Generally because of a lack of understanding and/or empathy between either party's views and someone skewing it out of proportion.
What an Idiot: Despite being a professional alien fighter by trade, Jack Harkness simply swaggers into Thames House and announces "no" to the 456 like a giant swaggering cock, then acts surprised when they kill everyone, including Ianto. One would think he would, I don't know, use science to figure out a weakness beforehand (which is kind of his qualification for the job)?