Cycling in pedestrian areas is already an offence
The lack of enforcement is perhaps why we have such a problem now.
I'm all for allowing assisted bikes anywhere a normal bike can go, but faster (actual) e-bikes; mopeds or motorcycles should be registered like an ICE version. They should stay on the road only. Given their ability to keep up with traffic, they're likely safer in that environment than an EAPC, and the rider would have some training too.
We haven't suddenly allowed motorbikes to race through pedestrianised areas and parks, or use pavements, and get onto trains. Why are we allowing electric bikes to do it?
If we don't sort it, as I said before, we'll end up with all battery powered bikes banned all over as a knee jerk reaction to what's going on.