From my experience in my country scooters are the problem, not bikes.

I have never needed to be watchful of a guy on a bike going fast by me, people on bikes don't go that fast in pedestrian zones.

But damn those people on scooters... Many go so insanely fast that I wouldn't mind a scooter ban in pedestrian areas.

Whenever I hear one behind me I always take care to move in straight line as I'm afraid if I move to the side the idiot will crash into me.

Whilst I much favour bikes over any other transportation means in urban zones, I think it sometimes makes sense to create pedestrian-only zones. Furthermore, cycling is overall good but not all cyclists are reasonable people - I've personally been involved in an accident caused by a cyclist (nothing broken but serious pain for a week or so) who would act as if the cycling lane belonged to him personally and it wasn't shared with pedestrians.

As -- I hope -- we move closer to banning personal cars in the cities, we should start looking closer into the dangerous behaviour of those using now-alternative modes of transportation.