'The general public in the UK are not permitted to own firearms. '

That is true - but in itself it isn't the really significant difference, which is that the general public in the UK have no desire to own firearms. Unfortunately that doesn't appear to be the case for a very sizeable proportion of the US public, and while that is true, making them illegal wouldn't solve the problem.

That means that it would be more sensible for them to go ahead with stuff like cutting out crazy weapons like assault rifles, and blocking loophole like sales at gun fairs etc.

And an idea that came up earlier in this thread would make sense too - follow up on the US Constitution gun amendment, and oblige gun owners to take part in regular training in a 'well regulated militia', as a prerequisite for owning a gun.   Their patriotic duty.