They always need to be in posession of the addresses, and deleting private keys completely makes no sense.
Im sure the exchange has the private key, but they don't usually hand those out to customers. If you mean something like bitpay, they do generate the addresses for you and handle the payment and deposit FIAT to your bank account. Coin payments and services like that allow you to accept different cryptos with easy and i guess to exchange all the income for bitcoin or choisen coin. They handle most of the payment technical side.