I'd add the importance of having a straight wrist and a curved thumb.
Thumb position is, vertically speaking, entirely a question of pragmatics; what are you doing? If you're doing a lot of bending or are fretting with the thumb, then thumb over makes all the sense in the world. Otherwise, generally centred.
Semi-disagree with you on using the thumb for support. It's a pivot point, yes, but there should be zero pressure, and should not be used for "squeezing" the neck to fret.
Also, rock players don't really know how to do it, but you can absolutely play on the upper frets without wrapping your hand around the body. You have two options, and can choose between either thumb position, like a cellist would, or placing your thumb on the lower side of the fingerboard for support.