I turned off Enhanced Safe Browsing for Google and tried again, but the window that automatically is generated while attempting to sign in to Apple Email with Google as my email account provider always opens with that Private Browser box.
To turn off the Enhanced Safe Browser in Google, I had to sign into my Google account and it had no problem allowing that, just opening email with google as the provider.
At this point, I must admit I'm thoroughly confused. Google is an email provider for Gmail, not Apple mail. So I've been assuming you're using a web browser to get your Google email?
It sounds as if you have two email accounts: (1) Google email account, and (2) Apple email account (likely an iCloud account).
If the above is true, then you should have the Apple Mail app (the icon in your taskbar that looks like an envelope) set up with 2 accounts...one for Google mail, and the other for Apple email. Please review this Apple Support article on how to set them up: Add email accounts in Mail on Mac - Apple Support
Once the Apple Mail app is set up with those two accounts, you'll be able to get both your Google mail and Apple mail with the same app, and no need to use a web browser.