Confusion about reduction counting vertex covers to counting cycle covers
Looks like the misunderstanding is this:

In the final reduction to (0,1)-permanent they are using modular
arithmetic, which breaks my argument.

Let $A$ be the original matrix and $B$ the (0,1) matrix.

Working modulo $n$, it might happen $perm(A)=0$ and perm $perm(B)=mn$.

Though equality holds modulo $n$, $B$ have cycle covers.



Haven't found the flaw in the question about maximum weighted
cycle cover, which doesn't appear to be affected by the above.