Is there a shortcut to Miller's algorithm?
Solution 1:
Every statement you've claimed is "obvious" is false at least some of the time.
First, $y-y_P$ does not have a zero of order $1$ at $P$ for some choices of $P$. For instance, for any point $P$ with horizontal tangent line, $y-y_P$ vanishes to order two at $P$. Next, $y-y_P$ has a pole of order three at $O$: the line $V(Y-y_PZ)$ has three points of intersection with your elliptic curve away from $Z=0$; the line $V(Z)$ has a triple intersection at $O$; thus their ratio has a triple pole at $O$.