I am flabberghasted that there seems to be a consensus that middle class parents 'know how to play' with their kids. In my experience, I've met many professional, educated parents who find playing with their children extremely difficult, perhaps because they share (consciously or unconsciously) the attitude that play is something for the thick girls at school. I've had many middle-class friends who speak of being 'mind numbingly bored' with 'entertaining' their baby. In clinic, I've met very well-to-do mums who find talking to their baby or toddler something very hard indeed, and are baffled by their child's behaviour and have no idea how to manage it.
People who are less well-off sometimes have far more social support within their wider family and community and have 'play without toys' (simple rhymes, games etc) modelled by those around them and are very, very interactive. It's such a fallacy to assume that money or background makes you better or less able to interact with your child. It also depends on the child you have: a child with additional needs, or who has a particularly difficult or demanding temperament, can stretch the limits of any parents' resources.
"I interact enough with my children at home" is a strange thing to say, I think. It's like saying that you don't/won't/can't talk to your partner when out for a meal because you 'talk to them enough at home'. It's a different setting with different interaction opportunities, more social opportunities etc.
No one should be stuck to their older baby or toddler in these groups, no, but it's not a babysitting group and you can watch from afar and have something to chat to them about after the group, too.
In our Children's Centre, the baby group (0-1) is more of a traditional coffee type group to build support networks for parents. However, the Stay and Play (for older babies/toddlers) is a free for all where, quite frankly, you need to supervise your child because (even if this was their remit) you couldn't expect one worker to supervise upwards of 25+ children running amok.
I find the snobbish attitudes on this thread appalling, tbh. Status does not necessarily make you a better (or even adequate) parent.