You've been mostly right - let's just get that out there in the open. Inversions are often used in passing chords, but in every modern style of music, jazz included, for piano-only playing or accompaniment, the majority of left hand action will still have the root of the chord as the lowest note.

Inversions; as with any other chord choice tool; serve to keep things interesting, but if all you have is non-root inversions in the left hand, things can get outright meandering and boring because part of the way we hear always frames higher notes to the root of the chord; it's what gets our hearing 'home'. Resolution to the root is always needed to some extent.

Have you ever studied the concept of 'pedal point' harmony? This is where there is emphasis on a note that is common to the family of chords in a song (though dissonant to a minority of the chords); very often in the bass register; or left hand. You can put the pedal up higher, but it's almost always the lowest note. Key however, is that there is almost always resolution and tension release to the key chord with lowest note root combination.

Great wikipedia article on it here across all kinds of music. I typically use it in instrumental rock compositions. It can give an epic-ness (if that is a word) to a chugging song.

https://en.wikipedia.org/wiki/Pedal_point

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