That is a fair question. I don’t think it sounds awful at all.
First off, part of this is the tradeoff; their future might not be grey and miserable if you lived, but might be if you didn’t. Now, you might not be super inclined to be self-sacrificing in such a case, but it’s usually a very persuasive argument, even though it isn’t really the best argument. To understand why, we have to talk a bit more generally about depression.
In general, the evidence is that suicidal people are objectively mistaken about their future. Which is to say: Among people who have tried to commit suicide, and are still alive a few years later, generally reported happiness is pretty high. Maybe not quite as high as the worldwide average, but they are consistently happy to be alive and glad they did not die. (“Consistently” doesn’t mean “absolutely 100%, no exceptions”, but it’s well over 70%, and I think over 90%. Someone who has stats handy could look it up.) So we generally conclude that suicidal ideations are not a result of a clear and accurate view of the future, but of something else.
Suicide is strongly associated with depression. And depression isn’t necessarily bad for your accuracy-of-thought, in general, but there is one very noticeable and characteristic failure mode: Your evaluation of benefits or rewards, particularly for yourself, becomes horribly inaccurate. This is why one of the most heavily-recommended things for depressed people is to try to help someone else. When you do things that benefit yourself, normally you feel like you’ve accomplished something. When you’re depressed, that short-circuits and doesn’t work. But! When you do things that benefit other people, depression is much less likely to intercept that and make you not feel it. So even though normally things that benefit you feel at least as good as things that benefit other people (by similar amounts), when you’re depressed, that’s not true.
Which means that if you tell depressed people that, no, really, the evidence is extremely strong that if they live five more years they will feel happier then than they believe is possible now, this will not seem persuasive even if they believe you. Because their evaluation of how nice it would be to be happy and carefree and alive comes back as “that doesn’t sound much better”, even though obviously it ought to be much better. But if you tell them that other people will be happier if they live than if they die, the evaluation-of-worth works properly.
So, just in terms of your own self-interest: You should stay alive because your chances of actually being happier and not hating yourself are actually really good, even though they don’t look that way. But since often it’s hard for people to believe that, people often provide advice focusing on a thing which is easier for depressed people to believe and assign value or worth to. That’s because your brain is lying to you about what you’re worth, and since we can’t just handwave that away (or we wouldn’t have to have the conversation at all), we have to present an argument in terms of claims your brain is less likely to lie to you about.
From a societal standpoint, of course, people might well care more about the many people who would be hurt by your death than they do about you personally, because there’s more of them than there are of you. And you would be totally within your rights to say “yeah, well, it’s not them living through this” to that. But conveniently, it really does turn out that you’re likely to end up happier too.