I don't see any major city doing that anytime soon. The grid is not ready, power plants are not ready, it would cost billions to rewire everything, especially in old European towns. And even more for germany since they're shutting down coal and nuclear right now. It's not a big issue though, we should focus on getting rid of personal cars and that's something Berlin is getting slightly better at by the day.
The UK National Grid says otherwise.
See https://www.carbonbrief.org/rise-uk-electric-vehicles-nation... for instance.
Secondly, grid has a lot of capacity. Its like motorways, most of the time its under utilised. Only during peak time that it is full capacity. Think of the grid in a similar way. Its mostly under utilised except for peak time. All we need to do is make use of it outside peak time (with batteries) and suddenly we have double or triple the capacity.
> All we need to do is make use of it outside peak time
Seems like a huge thing to overcome to me.