Algorithms for factorization using the decision version

Are there any practical algorithms for integer factorization that work by solving the decision version of the problem $\log N$ times to isolate a factor? Or is the decision version of only theoretical interest?

All of the following algorithms either find a complete factor or a complete non-trivial factorization:

1. Trial division.

2. Pollard rho.

3. Pollard $p-1$ and its variants, including the elliptic curve method (ECM).

4. Shanks' SQUFOF.