I speak in generalities because this is endemic of social issues and debates far beyond the realm of just this particular thread. I do not name names because while they have relevance to this particular conversation, the things you state are platitudes that are all over the internet and political conversations.

I reject the notion that simply having respect for people's mental well-being breeds hypersensitivity. It reeks of the American's right's shaky platform of "bootstraps" and comes precariously close to MRAs crying out that people are hypersensitive each time their sexist language gets called out (not the case here, to clarify). It's not hypersensitive when, as a gender, or a race, or a generally disadvantaged group--in this case people coping with illness--ask for a simple thing that can improve their lives, help them get well more easily, and has a net neutral impact on the presenter.

Yes. It does work that way. Quite often. I know a great many people that have them work that way and, as they are functioning, real human beings, they make the personal choice to read or not read, and having in mind what they may see does help them.

I have also seen TW's for PTSD. I know veterans personally who have sought help for their PTSD and in the process they brought up their finding of trigger warnings and, while it would not be helpful for him, it would be for some of his friends (he mostly had sleep triggers, while friends of his couldn't watch war movies).

Of course you can't find every trigger. That's not the intention, but we as people do our best to help make life easier for people. This is an especially simple thing to do that can be greatly helpful to people, yet people oppose anyway. I'm sure I can pull parallels to a great number of historical cases of that because it's just human nature.