There are some chords in jazz that can't be made without the thumb.

A full sounding ninth chord for example:

T3 M5 M5 I4 P6 R5

Oddly enough, barre chords are rarely used in jazz. Most of the chords are four and five notes which rely on muting strings with the fingers that are not wanted in the chord.

But the left hand thumb is used quite a bit.

Also, when playing ragtime type songs on a steel stringed accoutic guitar, holding down the F (on an F chord) on the sixth string with the thumb enables you to bend the second string on the third fret with the little finger. Can't do that with the thumb under the neck of the guitar easilly if at all.

However, sometimes for a clean single-string run it helps to put the pad of the left hand thumb under the guitar neck to free up the fingers.

I see the thumb being placed as being fluid depending on what is being executed.

I notice also that rock and blues guitarists tend to use the thumb as a leverage when bending notes by placing the neck of the guitar in the palm of the left hand.

Frank