We have several outstanding schools on our doorstep (zone 1), several parks, some with adventure playgrounds with playworkers, so kids over 7 can be left without hovering parent, it is diverse, we have lots of cheap restaurants around, travel is free for London children. You are never in a situation where the only childminder or cleaner in town has no vacancy. It is possible to walk or cycle to most places. Lots of baby and toddler groups, church-run, council-run, private, take your pick.
Kid's hobbies are well covered, several music schools and every imaginable sport in walking or 5-minute bus distance. 2 brownie and a scout pack close by. For teens, everything you could possibly want. Many of the hobbies very cheap and available for free if you are poor (inner city areas have all manner of funding for stuff like this). Kids who sing/ dance/ play an istrument get to perform at Shakespeare's Globe, the O2 and the Barbican not just the school hall. Routine school trips to Greenwich, the Tower and St Paul's Cathedral.
Drawbacks? Housing is eye-wateringly expensive of course and the kids will probably share a bedroom much longer than in the 'burbs. I imagine very few small-town doctor's or lawyer's kids share a room, in London, this is quite normal. But then the kids see it at their friends' houses and don't seem to be precious about it. Bunk beds are near-universal in zone 1 flats with children.
Would I live elsewhere? New York perhaps.