So yes, if you fund key 42 with no conditions, whoever signs first owns it. In the puzzle, the first signer already exists; a bot that shoves their transaction aside is just pickpocketing at mempool speed.
I do not agree, simply because it wasn't their key, so the ownership does not exist in the first place.
Are you aware that all puzzles with exposed public keys were signed before they got solved, since they had outgoing TXs? The sweeping TXs therefore classify as theft, according to your criteria.
Again:
not your key, not your coins. It is your key only when you actually create it yourself. Otherwise, it is not your key, it belongs to everyone and therefore anyone has the equal right to use it as they see fit.