As a general rule, I don't think it's right to hand a customer cleaning equipment!
However, from your posts, it sounds like the mess is being caused by things that haven't been purchased from the cafe. I think that makes a massive difference - a customer dropping the odd crumb from something bought on the premises is to be expected, and the need to lightly clean up after that person is built into the business model.
If, for example, you're bringing in baby pouches, getting sticky puree all over the high chair arms and on the floor, the cafe is doing you a favour in the first place by turning a blind eye to the external product - it's an ask too much to then expect them to dispose of the extra rubbish and clean up after the sticky residue of something they don't even sell. They're making a loss on this.
I'm sure all if not most of you have buggies with you and you're taking up more space than the typical customer - and I bet there are some people who are choosing to not eat in because they feel the group is dominating the cafe. (Doesn't matter who's in the group, a large party is always off-putting to some customers, which is fine if they're spending lots of money, but not if they're nursing a pot of tea for two hours.)
Basically, you need to be better customers.
When I take kids out, no one ever asks me to clean up after them, but I make a point of thanking the staff verbally, apologising for the chaos and also leaving them an excessive tip to compensate for the extra effort they've had to put in. I appreciate not everyone can afford to do this, but everyone can acknowledge to the staff that they've been a challenging customer, and if they don't have money to tip generously, they can offer to sweep up.
OP, if you had offered yourself, chances are the staff member would have turned you down. It sounds as if the brush was handed to you out of weeks of built up frustration. I don't think the staff member has covered themselves in glory, but equally, it doesn't sound like your group has either.
We're not all perfect, and sometimes the way to defuse the conflict is to simply own our imperfections. 'Thanks for taking such good care of us again - we tried to stop them, but we know the babies were messier than usual today. We appreciate you and your colleagues always making our weekly meet-up so pleasant.'