Change gmail password on Mac

Hmm, I just tried this myself and see what you are referring to.

I think you have two workarounds:

1. Delete your Gmail account from the Mail app, and then, re-add it back.
2. Create a new Outgoing Mail Account server for your existing Gmail account. To try this, from the Mail menu, select Mail > Settings > Accounts > Gmail > Server Settings.
   1. Click on the up/down arrows at the far right of the existing entry in the Outgoing Mail Account field.
   2. Select Edit SMTP Server List ...
   3. Click on the "+" button to create a new entry.
   4. Fill in required fields. This is where you can enter the Google app password.
   5. The Host Name would be the Gmail SMTP server.

I believe the first option will be the easiest. I have noticed that Google will send you a notice, via a pop-up window to enter your Google credentials and not even requiring you to use an app password.

Ref:

• Add or remove email accounts in Mail on Mac - Apple Support
• Mail Settings Lookup - Apple Support << To look up the Gmail servers that your Gmail email address uses. You would want the SMTP one for sub-step 5 above.