RSA algorithm is based upon factorization, reverse problem for which is in NP. If a solution's difficulty is a low degree polinom (3 or lower, but I might be wrong) it could be used to get private keys from public ones in a matter of days.

The proof itself would not give factorization algorithm.

Even if P = NP, prime factorization can be very time consuming to solve. The prime factorization problem is in the NP class, but we don't know if it is NP-hard.

Factorization is not NP, it's "easier"

It's definitely in the class, but it's undecided which subclass it belongs to.