Do you have the vertical grip on the stock?
My 2 cents. The thumb on the side of the stock is largely seen on the competition scene where the rifles don’t recoil, and pistol grips on chassis are very common.
In my opinion, for practical shooting, especially when more noise and more recoil occur, it’s easier to flinch, part of that flinch can be you grabbing the rifle with your hand wrapped around a grip.
The fix your attempting is to eliminate the thumb all together. This takes it out of the picture and you don’t get that response from your body.
The problem with this is not in recoil management, but rifle manipulation and control. If you place your thunder on the back of the grip, you retain more control over the rifle, and are able to make that grip if you need to manipulate the rifle.
Now with your thumb on the back of the grip, when you shoot you need to train yourself not to grab it. This is easier imo than with your hand fully wrapped around a pistol grip. Your not controlling recoil with your thumb at all, it shouldn’t be in the picture, just lightly on the stock. With a vertical grip you also gain the ability to pull that rifle into your body with your fingers. This is impossible for me without a vertical grip.
I don’t think thumb along the side is a reasonable way to be shooting a hunting rifle.