Hi Polly!
I may have a different take on this than others. I love playgroups. I have formed one of my own in each place we have lived. In fact, I started my first one when dd (now 5) was about 6 weeks old. I'm still in one to this day (now with dd3).
Which brings me to my point. I think most of the time, those playgroups were for me

. Especially that first one. And even up to age 2 or 3, kids mostly parallel play.
Sure, there are opportunities for the kiddos to learn about sharing and not hitting or pushing. But, as a full-time stay-at-home mom, what I found most valuable was the fact I had a regularly scheduled date for adult conversation. I've had the same playgroup now for 3 years, and through time and shared experiences those other mothers have grown to be my very close friends. And even though their kids may be different than my dd on the gifted scale, they and their children accept her in a way that others do not. They watched her develop. I didn't have to convince them she taught herself to read at the age of two. They saw for themselves how she changed from week to week.
This is the part of your post that drew my attention...
The playgroup organization assigned us to this one based on age, perhaps there's a chance of switching if I say my schedule's changed, perhaps not.
I think your problem may be two-fold. In going through a "playgroup organization" you may find yourself stuck with other mothers with whom you don't have much in common. And if your ds isn't enjoying it, either, it's pretty much a lose-lose.
Any chance of breaking off into a splinter group with the dc you do like (perhaps you enjoy this mom as well)? She may have other friends who are more "your people" (for lack of a better term). Or perhaps you have an already existing friendship with another mother that could transition into a playgroup? Even if the kids are different ages, I think it can work. In my experience, the more you enjoy it, the more the kids will, too.