For example, in an FPS you turn a corner, see an enemy and shoot them. But unbeknownst to the client a grenade exploded just beyond the corner killing the player before the shot got off.

Now we have to deal with the mess of the client thinking you just shot somebody while you were actually already dead. While it's true that this is "simply" a client-side issue, you'll hear endless complaints from players who, from their perspective, turned a corner and shot the enemy and then died to a grenade, but for some reason their shot didn't connect and to them it seems unfair.

Another scenario: An enemy client is approaching a corner. The server sends you their position because it estimates they'll continue moving beyond the corner. You see the enemy and shoot them. But then the enemy client's lagging input comes in and they actually stopped short of turning the corner! How do we reconcile the fact that you've seen and shot an enemy you shouldn't have been able to?