The reason the mayor is pushing this through over the Councils is that Councils don't want to upset car owners while the mayor needs some actual change to happen so the city can recover from all the damage done by car drivers.
I loved the classic complaints. "What about grannies, parents, and people with heavy shopping!?!?". I dunno, maybe we use some of the solutions other places have affected in the last decades for them? There's other forms of mobility other than driving your car and walking. Do what England does best: go overseas and steal stuff to use at home. You'll be fine. People survived before cars and can do so again.