Some private schools do and some don’t. Some state schools do and some don’t.
Sounds like your state school is a bit lacking honestly - DS is at a state primary and they all have the option of learning an instrument from Y2, all learn Spanish from reception, the ASC is all activity-based (DS is doing football and karate this term, did art club and yoga last year, there is also a cookery club, coding club, etc).
They go on trips each term. They do a Christmas play, act in class assembly in front of the parents each term, and do a summer dance show. This is in inner city Lambeth, and parents aren’t subsidising this (aside from the ASC which obviously we pay for, but it’s only £7.50 a session for 2.5hrs including tea).
So I can’t see why other schools couldn’t offer that as well if they wanted to.
The local secondaries also have plenty of afterschool sports and clubs, so I suppose it depends on what parental expectations are.