[Aguila Blanca]

I can't shoot with a "modern" high thumb or parallel thumb grip, but I don't think that matters for the purpose of this question. My view is that the support hand doesn't do any lateral pushing -- either left or right -- and should not be squeezing very hand hard. What the support hand should be doing is pulling back toward your body, while the strong hand is pushing the gun forward against the support hand.

I don't even understand your reference to "left to right squeezing is done with the left hand." WHAT left to right squeezing? There should not be any left to right anything, and most of the squeezing is done with the strong hand, to hold the gun steady.

Also, assuming that you are right-handed, I don't see how there can be a gap between the right hand and the right side of the grip. Any grip begins by holding your fingers and thumb out straight, forming a vee. The backstrap of the gun is then nestled into this vee so the barrel is parallel to the forearm. Then the three lower fingers (or just the middle and ring fingers if the grip is too short for the pinkie) are closed around the grip. The tips/ends of the fingers end up where they end up, but ... where does a gap on the right side of the grip come from?