I think it depends on whether you see the PC as a narrative character or as an avatar to immerse yourself in.
If they're a narrative character, then being tormented by the ghosts of their past, having to make tough decisions with their family/friends in peril, making sacrifices, and generally having bad stuff happen to them (that shines the spotlight on them, incidentally), are what you want to happen - that makes a good story.
But if you're looking at things from the perspective of the character, you see a thrilling tale of adversity and being forged by the hard lesson of life ... and you say "Screw that! I want the story where I achieve my goals, lose nobody important to me, and retire happily!" Stuff like having your family threatened by your archenemies is interesting ... when it happens to other people. But if you're looking to immerse yourself as a character, you don't want one with more problems than you have.
Of course, from the DM's perspective - the PCs are always narrative characters. And that's where you sometimes get tension - between the DMs idea of what would be an interesting story, and the players' concept of how they want their PCs lives to be.
Now personally, I'm somewhat in the middle - I won't hesitate from going into normal danger, such as overwhelming enemy force, deadly traps, and so forth. But I don't really find putting my character into certain kinds of stress fun.
I'm not saying I mind the occasional "evil forces are threatening your home town" plot. But always having to worry about your family/friends being used as leverage against you, or killed in revenge - just not my cup of tea. So I generally write backgrounds where the situation won't arise. And I wouldn't appreciate the DM changing that background to make it more "interesting" - those omissions aren't a mistake, they're a hint saying "I'm not going to enjoy that plot, try a different kind of threat".