I know this you can update manually using the Activity luncher but the main updater still doesn't work even after the last update and for the record I did contact the Xiaomi support they said :Try this fix. had somewhat the same issue. I was getting the old updater so I think the same fix will help you too.
did it work? i have the same problem. Even i got the google service update that notify me to restart to install the update 12.5.3 but i can't event press it to start installing. like my phone won't allow it.