Your private key is private - it is not meant to be shared with anyone, not even an exchange. An exchange should be able to facilitate a request but not have access to you private key. If an exchange knows your private key then why bother with keys? Why not let the exchange just provide you with an ID that they have created?
When you "move" your coins to an exchange, you aren't giving them your private key. Instead, you are paying them. At that point, the coins will be in the exchange's own private account. They will have their own internal ledger saying how many of those coins belong to you.
Why would you pay them? If you wish to
day quickly trade cryptos, they need to be on the exchange for quick access. This is the same as if you were day trading stock - it needs to be held in "street name" with a brokerage. It is too slow to mail paper stock certificates back and forth. Similarly, it is too slow and expensive to move coins to and from an exchange.