No that’s not how it works. When a transaction is submitted on the blockchain to withdraw funds from an address it needs to be signed by the private key and it exposes the full public key. A bot that would monitor such transactions would therefore see the public key. With just the public key you can’t create a valid signature, you still need the private key, however for this particular case, knowing the public key reduces the entropy of the puzzle by a factor of 2 (from 66 bits to 33 bits), so this puzzle was easier to solve for the bot knowing the public key published by the person who found the private key. This is very specific to this specific puzzle which had 66 bits of entropy. In general, bitcoin transactions have 256 bits of entropy.

If you have the private key you can send money as you see fit.

The purpose of the puzzle is to find the private key given only 75% of it.

Let’s imagine that takes 1 year to brute force the last 25%. But if you have the public key as well, it only takes 1 minute.

As soon as the coins were sent, the private key was known since it inherently revealed the public key.