The quantifiers matter

For any fixed $N$, a circuit that writes its already known factorization is easy to specify. That establishes only a statement with the weak quantifier pattern

It does not establish a polynomial-time length generator whose output handles every $N$ of that length.

Similarly, any finite collection of factorizations can be compiled into a finite reversible lookup table. Its gate count depends on the number of rows. Treating that row count as a constant while claiming an unbounded-input theorem would change the resource model.

The local circuit avoids a vacuous single-answer demonstration by handling every $2\le N<16$, but it does not pretend that this finite improvement proves the stronger uniform quantifier.