I updated to High Sierra on 8/3 and no problems until this morning when gmail stopped working exactly as your photo shows. If you don't see your email provider, tap Other to add your account manually. If you see Next, tap Next and wait for Mail to verify your account. Tap Add Account, then select your email provider.

If you only want to sync your Gmail with Mail app on your Mac computer, you can try to add other account without choosing the Google option in Internet Accounts. Here's how: Go to Settings > Mail, then tap Accounts. I used Airmail but that had worse problems on Sierra. I had the On My mac bug and I've had tons of freezing issues. I would mark spam as not spam and it would keep going to the spam box. It is advice, the Mac Mail app is outdated & has many bugs in Sierra. More specifically, some Gmail users have said that after installing the macOS 10.14.4, and launching the Mail app, some Gmail users are asked to authenticate their. It seems that the Mail App is not working as expected for some Gmail users. Several people are experiencing issues with Gmail (especially G Suite) email accounts in Mail on Mac.

Setting up mac mail with gmail