I FOUND THE SOLUTION.
People have mentioned it here, but with incomplete steps. So here is how to get it done:
1.) KNOW why this is happening in the first place: it's coz Google says your old mac and old mac OS have insufficient security. With that said, you can only update an old mac for about 7-10 years. Then you're SOL. Keep that in mind. Lucky for us old mac users (I'm running a 2011 MBPro with ElCapitan) -- we are still able to add new GMAIL accounts with a workaround.
2.) As mentioned here, open your GMAIL account preferences by signing into GMAIL by web browser and clicking on your avatar in the upper-right and selecting "Manage your Google Account". Then click on the "Security" tab on the left.
3.) Under "Signing in to Google" ENABLE 2-step verification. This alone will NOT fix it, but it WILL allow you use what Google calls "Less Secure Apps Access". If you notice in your account settings, Google automatically sets this to OFF -- essentially blocking access to any apps that Google dubs 'insufficient security'. Enabling 2-step verification will bring up a third line under "Signing in to Google" that says "App Passwords".
4.) Click on "App Passwords" and this will walk you through generating a password that you can use to manually add a GMAIL account with your old mac.
5.) Select and copy the generated password, go into Apple Mail preferences and select "other" when adding the new email account. Type in your gmail email address, and use the password you copied earlier, in the password field.
If this helps, PLEASE hit the kudos button to bring it to the top!
