I don’t understand what the issue is. I have a Gmail account, haven’t installed the Gmail app at all, but I get Gmail notifications and emails in the Gmail inbox in the Apple Mail app. Is there something I’m missing?
People are complaining that the Push function does not work with Gmail in their Apple Mail app, so there is a delay between when the email arrives on Gmail servers and then shows up in Apple Mail. Depending on the Fetch settings, this might take a while, as the default Automatic only works when your device is plugged in and on WiFi! The shortest Fetch setting is 15 minutes, so if you can wait that long, you would indeed still get your Gmails pulled in. Otherwise you would need to manually refresh for Gmail or any other services not matching Apple’s chosen Push protocol.