See, I like that idea. A series that balances their personal life alongside their crime fighting. Give it a slight slice of life vibe [You know like Friends TV show kinda vibe , highlighting their friendship/closeness] or have the series feature their romantic life just a bit more than the regular super hero comic.
If social media is a good metric then there seems to be a big interest in the superhero love lives.
I don't know what's on the stands from marvel but there's not any DC title that's current doing something like that.