Here is what Google Suggests to sync your phone:
Change date & time settings
You can try to force sync your account by manually resetting your time settings.
Open your device's Settings app Settings app.
Tap System and then Date & time.
Turn off Automatic date & time and Automatic time zone.
Manually change the date and time so that both are wrong.
Tap Home Key.
Open your device's Settings app and then System and then Date & time again.
Manually change the date and time so that both are right again.
Turn on Automatic date & time and Automatic time zone.
(Excluded 'deleting and re-adding your account' here as you already did this.)
Here are all possible options available by Google Support: Fix account sync issues
Google Play Store
In case you don't have Play Store App, you can download it from apkmirror , choose a version and install it after downloading.
You will be ask to enable 'Unknown Sources' to be able to install Google Play Store, follow this to enable:
Open Settings - Security Settings.
Scroll down to 'Unknown Sources' check the box.
After installing Google Play Store successfully, uncheck the 'Unknown Sources' box.
This will prevent everything (except the Play Store) from installing apps.