I believe the reason that England, for example, works well in banning handguns is that it's hard to get one at all, I'm guessing because those laws have been in place long enough for the guns not to proliferate and that it's geographically small enough to effectively control distribution

I'm not sure that that is true. We are always being told by the media here in the UK that is is very easy to buy a hand gun illegally for a few hundred pounds.

I imagine part of the reason we don't have a high level of gun crime in the UK is because the police don't normally carry them (they do have armed response units that can be called upon when the come up against a criminal with a gun). I'm not claiming that the US police could put away their guns and gun crime would stop, that is clearly nonsense.

It is difficult to see how the US can ever get from the situation it is in now with gun crime to a situation like we have in Europe. The US started with guns everywhere (at least that is how Hollywood tells us it was), which is very different to how Europe got started.