Solved Google Chrome An error occurred while checking for updates: Update check failed to start (error code 3: 0X80070002 -- system level).

OK, I found the culprit.
Manually checking for updates at chrome://components/ deletes the GoogleUpdater folder from C:\Program Files (x86)\Google\
The "Chrome Improved Recovery" component is probably causing it.

I found a Reddit thread about the same issue:
So the GoogleUpdaterService151.0.7910.0 service can't start because its executable path doesn't exist: C:\Program Files (x86)\Google\GoogleUpdater\151.0.7910.0\updater.exe

That's why Google Chrome can't update.
Interesting bug. I guess I need to stop visiting chrome://components/