I'm taking notes on chord inversions in common-practice period music and I noticed that the four categories of inversions (well, 6/4 chords at least) suggest 2 different purposes: a) making bass lines "nice" (i.e. arpeggiated, pedal, passing) and b) prolonging functions (i.e. pedal IV6/4 prolongs tonic, cadential I6/4 prolongs dominant). Is this observation accurate? Are there are other effects / motivations for inverted chords?
I would argue that there are other uses of inverted chords.
One other important usage is for dramatic reasons. A composer may set up a clear upcoming cadence, but at the moment our V chord resolves to tonic, the bass hops down to 3 and leads us to a I6 chord.
Although this can be understood as making a more melodic bass—and it could even be understood as creating a global tonic prolongation—I would argue that the main effect here is to dramatically deny the expected cadence, in turn strengthening the ensuing cadence that is almost certainly waiting around the corner.
Another reason is that, as with any different voicing, they just 'sound different' - while the basic chord 'name' might be the same, there's still a different flavour to the harmony.
An example: It's pointed out here "how much musical warmth and weight the low third in the bass provides" in Pearl Jam’s “Better Man”: