mgh = mv² is experimentally disproven in the ballistics lab experiments I mentioned.
"proportional to mv²" also means "proportional to 1/2 mv²" and "proportional to 123.45 mv²".
Assuming your gravitational and inertial masses are supposed to be the same, then 1/2 m v² is, by Noether's theorem, a consequence of conservation of energy. https://en.wikipedia.org/wiki/Noether%27s_theorem#Example_1:... .
Yes. This is my point. The error has never been found experimentally because everything eventually gets resolved to Newton's E = Fd. Dividing every result by 2 makes no difference because the proportions do not change.
I still don't see why gravitational mass is a problem but I'll keep looking into it.