Relation between key size and PRNG state size
Your intuition is correct.

While more bits of entropy is better when generating a key, attacking RSA through the key generation process is an independent attack of factoring the modulus, and is predicated on a different computational hardness assumption, search of the keyspace compared to the hidden subgroup problem for finite Abelian groups (for factoring or discrete logarithm.) Factoring at least seems subexponential but superpolynomial, so is significantly easier than search; Many search problems appear to be NP hard, with exponential time complexity in the average case.

The attack on the key generation process depends on the cryptographic strength of the PRNG, and is analogous to attacking a symmetric ciphers in many ways. Since the attacks against PRNG's are so similar to attacks against symmetric ciphers, you can use the guides at keylength for symmetric ciphers as guides to the minimum state for key generation as its entirely appropriate. If you're using RSA for key exchange, and your symmetric key length is smaller than the key length of your PRNG seed, they won't bother attacking the key generation process.

The question does highlight a very important concern about prudent cryptography however, and that is that key generation is highly vulnerable to attack by poor implementation. A cryptographically secure PRNG is essential if using a PRNG for key generation.