It seems that in whatever proof of the theorem that an elliptic curve can be put in Weierstrass form that you look at, the next step after getting an equation:
$$\alpha Y^2Z + a_1XY Z + a_3Y Z^2= \beta X^3+ a_2X^2Z + a_4XZ^2+ a_6Z^3$$
is to multiply through by $\beta^2/\alpha^3$ in order to make the linear change of variables $$\left(X\mapsto \frac\alpha\beta X, Y\mapsto \frac\alpha\beta Y\right)$$ to get rid of the leading coefficients. My question is: why are we restricted to linear change of variables that rescale $X$ and $Y$ in the same way (but still can send $Y\mapsto Y + mZ+nX$).
It is not that we can't rescale $X,Y$ independently, but rather that we don't want to rescale them by a square, or cubic root (as I suggested in my comment above $X'=\beta^{1/3}, Y'=\alpha^{1/2}Y$), which might not exist in the field $K$ over which the curve is defined.