Trust me it does. Wrist rotation is his primary movement, with him it’s more of a subtle gyration though, with his ring finger anchored on the pick guard and his pinky over the volume control, and not as visually obvious as when EVH does his fast eruption trem picking. He also doesn’t use a super firm grip on the pick, and is only holds the pick as tight as he absolutely needs to. When Yngwie tries to do all alternate picking across strings, which is something he doesn’t do very often, he uses more large forearm movement. This you can see in the 90’s young guitar videos and some things he put out I believe for guitar world in the earlier naughts.
Finger thumb movement on its own, is not a very efficient technique at all, and has very little power on its own. People who utilize it like yngwie or Eric Johnson are combining it with other motions.
Here look. His wrist is moving all over in this, also with some examples of forearm movement. His thumb and index are the joints that barely move that much:
What you may be confusing with thumb and index movement is likely his wrist movements while he is planted on the pick guard. Notice the core of his movement isn’t coming from his first knuckle joints, it’s coming from the joints that connect to the wrist.