I have 4 machines on Mojave, but on one of them (only), all my Gmail accounts are still f-ed up. This is in Mojave 10.14.6. They show as offline in Mail, I click to take them online, then it prompts me to authenticate in Safai, but the Google login page never opens in Safari (it will launch Safari if it's not open, but it doesn't direct to the login page). I deleted one account thinking I'd re-add it from scratch, but the same thing happens in System Prefs > Accounts as in Mail, prompt for Safari login that doesn't happen.
When this happened to me a few months ago, Apple support had me boot the device into safe mode, and then reboot. I was skeptical, but my broken Gmail accounts in the Mail app started working again after this process.