The current official definition of a planet is indeed problematic, mostly because it's a multi-part categorization in which several of the parts are not highly (or possibly at all) correlated. This matters bcause categorization is important in science, since it's important for people operating in the same fields to be able to use the same terminology among themselves this is true even when the categories are acknowledged as totally arbitrary as in the case of higher level Linnaean classification levels. Example: the family Formicidae is a completely arbitrary unit, but membership in the family means 'is an ant' while being in some closely related other group means 'not actually an ant' and these definitions are really important for myrmecologists because they need to all agree on what an ant is (and those various paleontological specimens at the margins induce fearsome arguments in the appropriate journals).

For astronomers working with non-stellar bodies it is important to decide on terminological categories for all the stuff in star systems that isn't a star. The actual physical properties of our universe only provide a few break points in this regard. One of them is 'is it big enough that it was ever on fire just a little bit,' another one is 'round yes/no?' Which is hydrostatic equilibrium. Generally everyone is content to say that any body large enough to have triggered some kind of internal fusion is a brown dwarf not a planet, and every object small enough that it's not in hydrostatic equilibrium isn't a planet. So far so good.

The problem comes when you try to divide up this population. There are two major issues. One is orbital - some of the bodies in hydrostatic equilibrium are orbiting other ones as moons. This is considered rather important, and it matters even from a purely compositional perspective because being a moon can have a rather drastic effect on a body's internal composition. Europa, notably, has a sub-surface ocean because of tidal heating from Jupiter that it probably wouldn't have if it were orbiting by its lonesome. Another it utilitarian. There are quite clearly sub-groups within the population of objects in our solar system (and probably all solar systems) that should exist as their own categories. There's the small rocky bodies, the small icy bodies, and the big gas giants just by composition, and there are also populations divided by position in terms of inner system, outer system, and really, really far out. The kicker being that you can only use the term 'planet' once.

Now, you can define planet as 'everything in hydrostatic equilibrium smaller than a brown dwarf.' The problem with that is the resulting list includes dozens of objects right now (including at least 19 'moons') and will probably balloon to the hundreds in due time. The IAU chose to bestow 'planet' on a far more restricted group, one supported more by linguistic history than anything else.