Your addresses (private-keys) are locally derived from the 12 word phrase that Exodus provides you when you backup your wallet. The email link (AFAIK) is also your word phrase but protected by both your passphrase and an Exodus 'key' (like explained in the page you linked). Your word phrase or private-keys isn't stored anywhere.
Sorry, little confused - Not sure if we're talking about different keys or not.
Are you saying without setting up the backup the wallet is functioning without private keys for deposited coins? - I'm curious as to how these keys are recovered with an online backup.