As far as I'm aware, each child got to choose something. So one chose horse riding and one chose dance. I don't know what the others chose and it is possible that the very youngest sibling was too little to actually do anything.
Horse riding was booked in blocks. I vaguely remember saying that the Mum could pay weekly, although everyone else was paying for four weeks in one go.
I effectively "underwrote" their horse-riding, should she ever have failed to pay up (which she didn't).
I don't know how the dance worked because none of mine did that but it was happening at the same time (by that I mean the same weeks) as horse-riding.
You're quite right, mine did not really meet the younger siblings as the age-gap happened to be too great. But they would have met other children at other things instead, just like mine would have met others at the other things they were doing.
Although horse-riding was quite a "big-ticket" thing (dance would have cost much less) there are lots of home-ed things that are free or almost free, such as park meets, co-ops and some study groups, arts & crafts, hall meets, so those things were also open to that family, too of course.
We took the "horseriding child" to a one or two other (one-off) things and they also attended a much much cheaper co-op/study group type of thing that we and some other families did every week.
Home-ed activities aren't permanent and static and guaranteed to be there for the next sibling.
Activities come and go (because the home-educator who is organising it stops wanting to do it, or the business or venue themselves choose to stop or it stops being viable because too many people fail to sign up for the next "block")
This is another reason why horse-riding in home-education isn't quite the same as the big financial commitment that signing up to for pony club would be. You can nearly always pull out when the next block comes around, if you need to, which just gives people a bit of reassurance. (There was also no kit to purchase either, the hat hire was included/free and everyone just wore suitable normal clothes)
Anyway, by the time the next sibling gets to the sort of age as the older sibling was, there would be something else on offer: perhaps climbing or trampolining.
There's also the option for any family to organise something themselves.
So if that family's younger sibling had wanted to do horseriding and it wasn't already running, they (the family) were free to organise it themselves and negotiate payment etc...
I'll also mention, because it seems kind of relevant: some of the loveliest acts of generosity and support that I've witnessed have been carried out by home-educators for other home-educators, in times of difficulty and that does seem to cross socio-economic boundaries at times.
I don't know for sure, it is very possible, likely even, that that family received help or support from other home-educators.