Google Domains DNS – MX Record Already In Use [Closed]

First, changing MX records wouldn't be done in Gmail, it would have to be done through your DNS provider (the service hosting your domain's zone file). That's most likely the place you registered your domain name but not necessarily, that depends on how you decided to set things up.

Second, even if you change the MX records of your domain and point them to Google's servers, Google wouldn't know what to do with those emails when it gets them. That's because they would be addressed to an email address they don't know about. The only way to have Google handle emails addressed using your own domain is to set up a Google Apps account with that domain.

If you don't want to pay for the Google Apps service [edit: now known as G Suite], the only workaround is to do what the other answer explains: 1) use an intermediate service to accepts incoming emails addressed to your domain and set it up to forward everything to your Gmail address, 2) set up your Gmail account to be able to send emails on behalf of another address

The documentation at the link that you provided says:

Your other email provider has to provide authenticated SMTP support for you to use this option. We'll use TLS by default, or SSL if you enable it. Many email services that provide POP or IMAP support also offer authenticated SMTP support, and you can likely find your SMTP server configuration instructions alongside information about POP or IMAP.

It seems that when sending an email from the alternate address gmail uses the third party provider's SMTP server, which would be listed as an authorised sender in the SPF record.