If you don't setup the backup & you're depositing coins, is Exodus operating without private keys?
Obviously not. The keys are already there when you first create your wallet. Otherwise, how would you have a deposit address?
That's why it's called a backup. It's only helping you make sure that you can create the same keys in the future if your PC breaks or something goes wrong.
Just understand one thing: ONLY YOU have access to your private-keys as they aren't stored anywhere online.