The server doesn't always decide. It's common for shooters to want to guarantee that if it
looks like you shot the other player on your screen, the game will react as if you shot the other player, even if the server would have considered it a miss due to timing differences. Those things can really only be done by trusting the client not to lie, which means verifying that the client hasn't been tampered with.
Plus, there's a lot of cases where the server sends some pieces of information to the client but the user isn't supposed to see it. The locations of the other players is a common example, where people modify their client to render outlines of players through walls. That's a huge advantage but doesn't even affect anything hit registration related.