Suppose we have an malicious agent Bob with account balance of 100 units trying to double spend a coin by broadcasting two messages simultaneously:
Bob sends Alice 100 units
Bob sends Emily 100 units
Now some nodes in the transaction will receive message A first, and others B first. If a node receives both messages, it will only validate the first message (since Bob has only 100 units in his account).
Theoretically, we can figure out how many nodes consider each message to be valid, and add the transaction with the most number of validators to the blockchain.
So why exactly do we need PoW in such a scenario?