It is strictly necessary that the longest chain is always considered the valid one. Nodes that were present may remember that one branch was there first and got replaced by another, but there would be no way for them to convince those who were not present of this. We can't have subfactions of nodes that cling to one branch that they think was first, others that saw another branch first, and others that joined later and never saw what happened. The CPU power proof-of-work vote must have the final say. The only way for everyone to stay on the same page is to believe that the longest chain is always the valid one, no matter what.

We're not "on the lookout" for double spends to sound the alarm and catch the cheater. We merely adjudicate which one of the spends is valid. Receivers of transactions must wait a few blocks to make sure that resolution has had time to complete. Would be cheaters can try and simultaneously double-spend all they want, and all they accomplish is that within a few blocks, one of the spends becomes valid and the others become invalid. Any later double-spends are immediately rejected once there's already a spend in the main chain.