Well, when it is clogged, I have tried filling the bowl to the brim
(which should be as good as a bucket full, right?). Some of the time
the extra head of pressure clears it but most of the time I still need
the trusty plunger.

If you are saying that I should try the bucket *before* I get a clog,
how will I know whether the bucket helped or whether it would not have
clogged anyway?

Ok, I think you have successfully demonstrated that the problem
is *NOT* insufficient flow from the tank to the bowl.

Which means that you have to drain and pull the toilet
itself, and check IT for obstructions, and look down the
waste pipe and see if there's either standing water or
an obstruction there. If that in turn doesn't reveal
the source of the problem, have someone come and
snake the main drain.