I think they need only to check if the coins reached on the deposit address on both chains. Then these coins are safe. It doesn't matter then which chain will win since they have the bitcoins on both chains.
Huh?
You are not making sense. An exchange is going to on one fork or the other (not both unless they want to be giving away fiat).
If your exchange decides on Fork A and then Fork B finally wins out then that exchange will go "belly up" (as all the coins it has had deposited would be unsalable).
You are right, they will chose one chain. But that doesn't say that they can check incoming deposits against the deposit on the other chain. When they received the bitcoins not only on their chosen chain but on the fork too then they can accept that deposit because it is riskless.
Why should they not monitor the other chain and instead close down completely? That makes no sense businesswise.