The HE group I help to run is led by the children. Most of the families who attend take an autonomous approach, so suddenly having lots of structured activities wouldnt' really work. Having said that, there are other more formal groups going on that most of the local children attend too. It's just that the group we run is meant for socialising, not for learning.
We have paper and pens on a table, or playdough - something for the little ones. And usually a parent or two brings along an activity but the children never spend much time on them and usually just relish in the big chunk of time they have to play with lots of their friends in the outside.
Most areas seem to set up groups that work for the families who want to meet up in that area. Some seem to have very structured groups, others more flexible ones. If the group you found doesn't meet your needs, then don't go, or go but set up one of your own that does work.
The thing is that most people who run groups do so out of the goodness of their hearts so they tend to set up groups that they think, primarily, their own children will benefit from, and then hope that other families will also find benefit in it. In our case, we have 7-15 families every session, so clearly it meets a lot of parents' needs. Some parents say they'd like more activities, but I say 'fine, run one then!', and usually they do!
Other parents who have specific skills sometimes offer to run something in a 'skill-sharing' way. So one local mum who is a musician runs a singing group weekly. A group of us asked a local French man to run a small french club weekly. Another dad who is a PE teacher runs a weekly PE group. My skills aren't something I can use to do that sort of thing, but I'm good at organising so I run the group I run, and I also set up museum visits etc.