The hard fact is that enough people can, if only judging by how over-subscribed many of these amenities are. Whether or not this is sustainable remains to be seen - I certainly have bones to pick with the IMHO unwise government policy decisions that are pricing essential service workers out of the area - London can not function on bankers, lawyers and techies alone.
There are a fair few free amenities in London - museums, art galleries, concerts sometimes. The parks and gardens are good too.
I thought it was funny how in London museums mostly have free entrance but you have to pay to visit some churches (St Paul's & Westminster Abbey).