For us, I don't think it's a case of helicoptoring vs. free range...I think it's a case of we are both full time working parents, and we want to spend the time we're not working with our kid. I would never turn down a playdate invitation for my kid, but at the same time why would I seek out opportunities to dump her on another parent, or send her off to play alone...when my time with her is already limited?
As far as her preferences, she'd spend 25 hours a day with her parents if she could. i'm sure that won't be the case when she hits teenhood, but for now I'm happy to oblige as much as possible...