Why cant you use randomness to seed more randomness?
Obviously you can, that's what any (Cryptographically Secure) Pseudo Random Number Generator or (CS)PRNG does after all.
Could the resulting hash be considered "perfectly random data" as well? I am assuming the answer is "no", but don't know why.
"Perfect random data" does not really have a formal definition. However, let's assume that we want to use this data as a key stream in an OTP, a use case that is more or less implied by using "perfect" in the term. In that case an attacker should not be able to find out any information about the values of the plaintext (bits) nor by extension of the key stream.
It is easy to see that the random data created by the hash function doesn't have this property: only specific messages (say of the same size as the hash output) are possible as the set of ciphertexts are limited by the amount of hash values available - 256 in this case. So no, in that sense the hash output certainly is not perfectly random.
The above assumes that every bit within the hash result should be considered random, and this assumption would generally be true for most use cases. However, the hashes are still perfectly random within the group created by the 256 unique hash values. There is a known 1:1 mapping between "seed" and hash output in this case.
This won't hold for any seed size due to overlapping hash output values (the pigeonhole principle). It should not be possible to generate a hash collision so this is of no practical concern. However, for perfect security we're not limited to practical concerns.
So the answer relies on the output domain; the answer is no-but-maybe-yes. This shows that it is impossible to have a rigorous answer without a well defined question.