Things like this urk me for a lot of the same reasons you had brought up. DMing is telling a story and while so much of table top is improv there are still moments that if you took them away from a campaign would cause the whole thing to fall apart. Players are meant to try to outsmart the DM in a way, use your gear and abilities to do things they didn't expect from you and you're having a great time. Wrenches get thrown into the cogs every second but giving anyone carte blanche to throw that big a wrench into the works is not a good idea.
If a single person is uncomfortable or unwilling to go along with something there is the option to just speak up or walk away whether in character or in the real world. If it's a group of friends I'd hope you'd found people who mesh well enough with you to accommodate you, and if it's at a game shop or con with people you'd never met then most people don't want to have "I made someone cry." as part of their reputation.
Even a situation that is uncomfortable or unpleasant doesn't mean it's harmful to you. Recently my grandmother died and my job gave me a few days off to help get things in order and grieve. I do much better when I have something to do in any situation but I needed something even more then. Recently a friend had paid me back with a game called Va-11 Hall-A. It wasn't my usual type of game but I was interested in the setting enough and started playing it. The entire time I played I felt too like the main character. At a point I would pay more attention to who spoke next and try to guess what she would say, occasionally getting something dead on aside from the wording.
Roughly halfway through the story it's revealed that a character from the protagonist's past has died and a relative has come blaming the character. Lliterally the day after my grandmother's death I'm faced with a story where a person's death is a crucial plot point. I had an x-card, I could've turned the game off and never looked back. But I didn't. It kept going and I kept seeing more of what I was feeling coming through in the story. It was a therapeutic look at myself and the way I was dealing with my emotions.
It wouldn't be an understatement to say that the game had changed me a bit. Running and hiding from it when it comes up in a controlled way would have been me robbing myself of personal growth. I doubt that many people can craft situations that would be so useful but it's not something you can tell from the moment you see things starting to veer in a direction. If a game isn't the place to look at and deal with what mentally disturbs you then where is?
This sort of is the "X-Card before the X-Card was a thing" … next to "Talk about it".
You always have the option to leave, but if you're having fun every other way, and the thing you're uncomfortable with doesn't seem to be a major thing, skipping it, to continue having fun, and seeing the other (non-uncomfortable) story end would be better than leaving, wouldn't it?
In RPGs there is more stuff going on, and the players (which the GM is one of) are writing a story together. Saying "Please don't make emotional deaths a thing" or raising the X-Card in case of an emotional death on the horizon (which should be the same thing) is better than making someone cry at a convention.
Even if it were for me to "tough it out" to advance as a person, uncertain of whether I could take it, or have to break down, crying after ten minutes, I don't want to do that with strangers at a convention, but at my home, ideally with friends.
I'm really unsure as to what exactly you're trying to say other than don't go into certain places in your stories. Certainly when you're doing a one-off with a group of strangers you don't go to places where things like character deaths, genocides, economic strife, ect would be used but that's more because of a time constraint. You have maybe 6 hours max with these people and lots of that is going to be slowed down by things like combat and side distractions. Those things being used as story points can't function in such a small amount of time and the story told suffers because it can't have its payoff.
I don't understand why anyone would be against something like an emotional death in a story long enough for it to be effective. It's the only way to be respectful to death in a story. If there is a death and it's someone of importance would you rather their death be unemotional? Would you rather they never die? If you've asked for there to be no deaths then you know there will be no deaths. If you know no one will ever die then anytime they're in a life threatening situation you don't have to worry and all the tension is gone. Balancing the possibility of death is a big part of why designing combat encounters is so difficult.
I think the X card is a bad idea because it's like codifying "don't be a dick" into the law. If you want to have fun and play the game you'll be accommodating. It's an insult to any decent DM or player to suggest that someone needs to have one. It pulls me out of my character to think that maybe someone could just say they don't want something and poof things are different. Everyone is making a story together, sure, but no one can be entirely in control. Not even the DM should have the power to just zap something out of existence. They have to own mistakes and bad die rolls just like the players do.
If there is a situation that is disturbing, uncomfortable, or unfun for a player then there is probably some point to it. Maybe there's a villain being set up, every good story needs an antagonist. Maybe it's a consequence of the players' actions, over throwing a leader causes lots of problems. Maybe it doesn't even involve the players but is to set motivations for an NPC, vengeance (something that would require a death to be emotional to atleast on person) is a way to give a goal. Stories that are free to explore things that aren't the normal standard subjects are the stories that are the most interesting and impactful ones. That freedom is what sets table-top apart from every other form of gaming. Allowing someone to stop that from happening is asking for worse campaigns.
If you want to use it and everyone else you're playing with has agreed to use it then go right ahead. To try and make it a rule is where the problem lies. Let anyone make any story they want because trying to stop it is impossible. If you'd rather not participate in something specific then just don't participate. Don't stop everyone else who is alright with it from going down that path.