You can use you own server with eg. Postfix and Dovecot to do this, either with you own client or use Gmail to read and send the mail. The downside with using just Gmail is that mail will be delayed as Gmail doesn't check inboxes that often. Mail sent from can be stripped for any Gmail headers when being sent out, if you want.

I've used Gmail like this for maybe seven years to avoid gsuite and still have custom domains when using Gmail. The delay seems to vary based on how often there is new mail.

You could use a webmail solution or K9 mail to get mail instantly.