Vaalingrade
Legend
This is why I love HERO, as Disadvantages are configurable versions of this.One thing that may help with that? Let the players set limits.
I did this with a Sci-Fi game a friend ran. I wanted to play the reluctant hero, a simple engineer trying to get home to his wife, whom he had wed before the company sent him on a multi-year mining tour. I told the GM flat out, I was not interested in any stories that involved his loving wife leaving him for another man because he'd been gone, or anything of the sort. That wasn't the story I was interested in. She ended up terrorized by a gang because his entire homeworld had turned into a mafia world for some reason, but the GM respected my call on who this character was.
A lot of players avoid connections because they not only know the DM will use them as "knives" as one of my former DMs put it, but because they don't trust the nature of the knife. But if you give some of the control back to them, and respect that control, then they may be more receptive. Something like "Hey, I want to tie you guys into the world more deeply and create some potentially interesting drama. Can you guys give me three 'knives' I can use against your characters? I'll let you tell me what is off-limits."
And sure, you may have that player who is like "Here is my sister, I don't anything bad to happen to her ever." But you can still work in some plot points with that. Like, she finds a genie spirit and gets three wishes, and that starts some chaos. Nothing bad is happening to her, but she becomes a focus of a story. But, in my experience, most people are okay with "This is the person I want to save later as a big hero moment" so they are fine with them being put into danger, but they wouldn't be fine with their childhood friend secretly being a devil worshipper and having to fight them, because you thought it was more dramatic than her being kidnapped.
For example, you have a rival and you can decide if they're a romantic rival or professional or whatever, whether you're actually trying to kill each other or if they're more or less powerful than you. So you can have a nuisance you kick around like shocker, or a deadly crime boss out to end you like Kingpin.
You also have Dependent NPCs; people your character cares about. You get to dial in how often they're in danger and how bad the danger is based on how many points you get for them. And then if you don't want them to be in any danger or to be helpful, they can be Contact instead and the ST isn't supposed to screw with that explicitly by the rules.