Gmail – How to remove all traces of the original Gmail address from a “send mail as” alias

Background: Forwarding custom domain's email to and from Gmail

I've got a personal Gmail account that I've had for many years at example@gmail.com. Of course, the address isn't “example”, it's something I set up on a whim years ago. Now, I've got myname@mydomain.com via email forwarding in Namecheap, and would like to use that for all my future professional and public interactions.

What I've done so far: Email forwarding and Send Mail As

I used Namecheap's domain configuration for “All Host Records” to use their Free Email Forwarding option, and added byname as the User Name and example@gmail.com as the Forwarded To address. I'd like to use Namecheap's email forwarding, as it's super easy, but I'm open to running my own mail server, as long as it, too, forwards mail to and from Gmail.

Then, I set up the example@gmail.com account to send mail as myname@mydomain.com through Settings → Accounts and Import → Send Mail As, and I've tried this with Treat as an alias both set and cleared. The mail gets delivered, and shows that the sender is the myname@mydomain.com address, and clicking “reply” populates this address in the “to” field as it should. This is all well and good for 99% of the population.

What's not working

Unfortunately, showing the original message from within Gmail reveals that the Sender, Return-Path, Received-SPF, and Authentication-Results all contain example@gmail.com.

Why is this information leaked? Is this something I should be concerned about? Finally, and most importantly, how can I send mail from my new address and use Gmail as the mail server without leaking the Gmail address?