Question about SHA-2 (and potentially similar hashing algorithms)
Would it not be better to use a set of less predictable values?


Actually, there aren't any "unpredictable values".

SHA-2 is a hash function (actually, it's a family of 6 hash functions; we'll ignore that point for now); a hash function is designed so that anyone can compute it.

One use of a hash function is during signature generation and verification; typically, the signer takes the message, and hashes it, and then applies the signature operation to the hash.  Then, the verifier takes the message, hashes it, and then verifies the hash against the signature.  If the verifier could not compute this hash, he could not verify that the signature corresponded to this message.

Because of this, the entire specification of SHA-2 must be made public, so that the verifier (and anyone else) can compute it.  That is, any internal table values must also be public, because they are needed to compute the hash function.

Because we're going to tell everyone what the values are, they aren't, in any sense, unpredictable.