The public key is already known information, but that is not enough to crack the private key, and if someone get's access to the private key first they need to spend the coins before be able to double spend it. But looks like you are lost in the topic, what op means is.

Let's say someone finds the private key from puzzles 66 and 67, if they spend the coins and make public the private key, som users could use that private key to make a double spend from the same coins and if they use bigger fees then they could steal those coins. But the way to avoid that is to not make public the private key until it has more than 1 confirmation.