The way you speak in absolutes and hyperbole suggests that maybe you've already dug your heels in on this issue, and that your questions are rhetorical. (But still I persist.)
No, a puppy doesn't die every time somebody refuses to use 'they' in the singular.
I'll admit the singular use feels a little grammatically awkward to me every time I see/use it. But you know what? It doesn't actually make my life any worse, either.
And there's a whole segment of our population...you know, fellow human beings...who feel that the lack of an appropriate pronoun to describe themselves contributes to the isolation and bigotry they've suffered their whole lives. They (plural) feel strongly that it reinforces the popular misconception that gender is purely binary, that you're either completely one or the other, and that by not fitting into that false dichotomy it opens them up to vicious discrimination. Structurally it's really no different from nasty racial epithets: labeling propagates vicious discrimination.
And even if that's false, and there's no correlation/effect at all, there is absolutely no denying that our society craps on those people. If they believe that this small sacrifice will help alleviate that, then fine.
As somebody at the extreme end of privilege on just about every metric, the least* I can do is tweak my grammar a little bit to accommodate a group that doesn't have my advantages. And those who begrudge them that are a far, far bigger problem in the world than a little bit of grammatical drift in our language.
*Oh, and I can vote! And make campaign contributions! Yay!