The thumb draw is used due to pinching of the string. With such short bow lengths, pulling the string back to full draw creates an severe acute angle near the nock. This means using traditional methods like the three finger 2 under draw will not have a clean release and your hands will be pinched.

That's one of the characteristics/advantages to it but there are too many longer bows being shot with the thumb to say that's why they do it. [Ishi] shot with the thumb, the Japanese with their 7ft bows shoot with the thumb, Mongols/Manchus shoot a bow practically as long as the longbow with their thumb.

The issue with shelf side is the the thumb draw produces torque on the string causing arrows to want to rotate. This means if you draw with your right arm, the arrow will try to rotate counter-clockwise so you need to use the right side of the shelf.

Not true. Everything is quicker and more convenient if you put it on the right side but it isn't necessary. There is practically no torque on the string, what archers are doing is putting torque on the arrow directly with their index finger.

A lot of Mongols shoot thumb ring with arrow on the inside of the bow. They do this because they want to aim by gap shooting. Paradox is inverted but they just need to spine their arrows a bit differently and they're fine. Their arrows aren't falling off their bows.

A correction to the op is that inertia not momentum is what he's trying to explain. Both are based on mass, but momentum is proportional to velocity. This is true from the Newtonian perspective mind you.

I don't know what you're trying to correct.. Momentum is what determines how far a bow would twist before stopping. Mass affects how much the bow gets decelerated by friction between the hand and bow and the velocity is how much speed the bow has to lose before coming to a stop. Technically angular momentum, torques and angular velocity but whatever.