We already have the word "satellite" for "things that go around other things" right? I think "moon" is just "satellite of a planet" for convenience in discussing that subset.
> Why is having a lot of planets a problem in the first place?
I think keeping the number manageable is explicitly something we keep around to help kids grasp the main entities in the solar system. If we just said "there are 235 planets" it would be silly to try to teach them all, so we'd probably just settle for "The top 10 biggest planets" or something. Having a definition instead of a number to bound the set isn't much less arbitrary than teaching the "top 10," but since the long tail clearly starts after #8, "Top 8" would be the only guaranteed stable set to give special treatment to, which is what we've arrived at with the official definition.