Did the puzzle creator specify what "real solving" means?
No.
A grinding solver broadcasting a vulnerable transaction in the public doesn't own the coins until the transaction is actually confirmed. Am I wrong with this? I'd love to hear why, seriously!
Then, it is similar to this puzzle:
https://bitcointalk.org/index.php?topic=293382.0Exposing a vulnerable public key in public mempools opens the opportunity to use faster methods than brute-force grinding of the private key. If a real grinding solver ignores this, whoes fault is this, seriously?
Of course it is solver's fault. However, there are ways to do it in trustless way, they are just not used, because trusting centralized pools is sufficient in practice, at least for now. But it can be done differently, if needed.
What exactly is wrong or unethical or whatnot to use publicly available data to find a private key faster that allows you to sign a transaction to move coins that are "controlled" by such a vulnerable low entropy private key?
1. The puzzle creator can always sweep all coins, at any time. There are other puzzles, where it is not the case.
2. People have to trust, that private keys have N leading zero bits, and that someone really solved it, and the creator didn't sweep it just to raise some panic. If DLEQ proofs would be available, or other similar proofs, then everyone could validate it, without trusting anyone.
3. When it comes to hashed puzzles, there is a trustless way to prove, that N-bit hashed keys are no longer safe. It is currently done by using vanity addresses, but it could be potentially improved, by wrapping it into Script somehow (I don't know yet, how exactly, because this feature is not supported directly; but technically, it can be done).
How do we define "ownership" of coins?
By looking at the Script. There is nothing else. If someone can break ECDSA, then that person owns almost all coins, which have known public keys (almost, because it is possible to make a Script, where knowing the private key is not enough to move it).