Why is it that when using either a ball mill or a radius end mill, good
finishes can only be achieved by climb milling?

At a guess, it's because conventional milling requires cutting into
stock from the thin side of the removed material, making it more
likely that the cutter will deflect, smear, or skip before it has
enough tooth load to dig in and cut, but a round endmill exagerates
this even further because you're cutting an even thinner chip down on
the radius, and that switching to climb milling loads the teeth enough
to put the cutter back into a reasonable cutting depth.

--Glenn Lyford