This isn’t a mistake per se, it’s how that class of hash functions—and really, almost every hash function ever—is implemented. It’s called the Merkle-Damgård construction. It adds some very good properties and is the basis for how hash functions can be used in hash tree constructions and such.

But proving that the input state is evenly mixed among the output state is THE hard thing to prove (the hash function equivalent of the difficulty of factoring integers), so for the sake of ecosystem diversity NIST chose a hash function based on different principles for SHA-3. It’s not a criticism of SHA-2 that the difference was called out.

The constants are the fractional bits of of successive cube roots. This is effectively a nothing-up-my-sleeve random number selection. If there are problems with this, that in itself would be a serious cryptographic result.