My thumb sometimes hooks right over the top of the fretboard and frets strings, it makes some chords easier e.g.
5
6
5
0
0
5 (thumb)

Generally my thumb doesn't move much but it does (without deliberation) move and help me position my fingers when I make a big sweep along the length of the fretboard.
Don't get hung up about your thumbs,they're your friends Personally I use both thumbs when I play.