I still stay with my older child at playdates unless I know the hosting parent well because quite frankly my kid can be a pain in the arse and I don't want to leave him with anyone who isn't capable and happy to deal with that. I have started leaving him with parents I know well who are able to squish him without too much bother with the caveat "Call me if he's being a nuisance and I'll be round asap to pick him up".
Personally I don't find playdates particularly enjoyable as a parent and I'm certainly not doing them to advance my social life. I smile politely, make friendly conversation and am grateful for a cup of coffee. Hopefully I have sufficient social skills to make it seem like I am enjoying the company of the other adult(s) and more than occasionally, since the parents of my children's friends are mostly very nice people (it's just that I'm naturally introverted), I do find myself enjoying their conversation and having actual fun.
My DC has been at school for a number of years now, we have attended and hosted countless class parties, attended school plays, other school events, volunteered for school fairs and class trips and attended and helped with fundraising events. I know most of the other mums, and quite a few of the dads, very well by this stage. We help each other, pick up each other' kids, give lifts, share information.
I know it's usual on MN to deride other parents outside the school dates as "school mums" and not important and not real friends and all that, but at the stage we're at in life with our kids, these people are our "peer group" in much the same way as their children are our DC's peer group. Like all peer groups, there are those we like more and those we like less but we share a bond with all of them.
Let's put it this way - my "real" friends (from BC ("before children")) aren't queuing up to drop my kid home when I'm running late or trek out to some godforsaken soft play to celebrate his birthday on a Saturday morning.
Kids are fickle and our children's friendships are up-and-down. But regardless of who's playing with who that week, there are some parents who are reliable, always turn up and offer help if you need it. And even if our kids aren't presently best buddies, it's always nice to have a coffee and a chat with those parents.