What I've seen is that, to transfer your coins from your exchange wallet to your software wallet, you use your software wallet's public key, and to transfer funds from your software wallet to your exchange's wallet, you use the exchange wallet's public key.
For making any bitcoin transaction, whether it's a transaction to a wallet or an exchange, you need the recipient's address.
Note that although your address can be derived from your public key, but you always need to enter the address when making a transaction.
Basically meaning you can keep your funds safe by never using your software wallet's private key. Do you need both?
Your private key is used only when you want to make a transaction from a non-custodial wallet.
As long as you don't intend to spend bitcoin from your non-custodial wallet, you don't need your private key.