Let's assume the mempool is low when you do the transaction for the puzzle. If I send the coins with a fee of .1BTC and someone else finds the private key 30 seconds later and sends the transaction with a 1BTC fee, would they both be in the same block and the first transaction would win because it has an earlier timestamp, or would that not matter?
Double spending in Bitcoin is an inevitable feature of the network, regardless of whether it is legitimate or not.
So how do you solve the problem of a conflicting transaction? well, through the confirmations of the miners. And how can we "win" a conflicting transaction? increasing the rate...
In fact, exchanges usually carry out these transactions to reverse a transaction in which the transaction fee was low and they need to make the transaction faster or even unblock a stuck transaction generating a double spend this time with a higher fee.
So this is part of the bitcoin protocol and there is nothing to do. If someone obtains your private address and makes a transaction before it is confirmed, both transactions will "fight" to form a block first, regardless of which was issued first.
It turns out you need to win the lottery 2 times)) first win the private key and then also so that your transaction is included in the block. But what if you choose a moment when the network is overloaded and make a transaction with a commission of 100 million sat? Will this increase the chances or does it not matter?