It's just an observation that kids often ask for playdates but mum decides which ones will be facilitated and very often it's only the ones where the mother is essentially facilitating playdates for herself, because she wants to be friends with the mother herself, because of the jobs, houses, class and other things to advance her social climbing.
Genuine friendships aren't supported, the kids needs aren't met, kids get message they aren't important. It just routinely seems to happen all around.
Assuming that the children know each other from school, then surely they're all friends anyway regardless of how many playdates are organised outside class time? The mothers aren't forcing them to play with kids they hate. You wrote yourself that it's a requested play date and the mums are simply deciding which invitations to accept or reciprocate.
Some degree of social engineering is always going to happen when kids are too young to make their own decisions. You seem to assume the only single purpose is social climbing but I think that many parents instinctively prefer their children to have friends who are happy, innocent and have all of their needs met at home. This is absolutely NOT saying that all kids from less well-off families are problematic. However in the absence of knowing more about other families aside from a few glimpses at the school gates, most parents will gravitate towards the families which have the most obvious class signifiers.
A wealthy family with both parents happily married and lots of support from grandparents and staff is most likely going to produce happier children. A family living on the poverty line with lots of trauma and abuse is an environment that may not be conducive to producing happy kids. A lot of parents may not be interested in social climbing but they are picky about who their children spend time with outside school and if that environment is safe for them.
I would argue that the "popular mums" are not the ones who have most money (although this frequently overlaps) but the ones who appear most trustworthy and eager to host their children's friends. I personally hate tracking down the "I'm too good for the school Whatsapp chat" type of mums. DD may request a playdate but it's a red flag for me if a mum can't even be arsed to let the parents of their children's friends have a way of contacting them.