This reason this isn’t straightforward is because with G Suite all kinds of organizational settings are in play and many of the switches needed are in the wrong state by default. As the admin of the organization you have set these up properly.

First off, granting access to the account would be the straightforward, native way to do it. Unfortunately that’s only possible within the organization so it’s usually not an option (nevertheless here’s how to turn it on). That leaves SMTP for sending and POP or forwarding for receiving. Forwarding is instant but the recipient will be your main account (instead of the G Suite one) and that will be the default sender address when answering as well. Whereas with POP the recipient is the G Suite address and Gmail can be configured to use it as sender when replying but the sync can be very infrequent (up to one hour delay, probably the less the more frequently you get mail). There’s no official way to speed it up but that algorithm allows for a nice little hack.

Keep in mind that all changes can take some time to be active. In one case I had close and reopen the setup window but it might’ve just been a timing thing. Also got locked out of my account once because of “suspicious activity”. Be sure to have a recovery email or phone number set!