WhatsApp chat backup in Android backup on Google Drive
Is there any way to retrieve my WhatsApp chats from this phone backup on Google Drive?

I don't quite understand, if you want to backup from Google Drive, if it's been set up, then it should be automatic. Check your google drive, top left menu, backups, if there exists a whatsapp file. If not, you don't have a backup in GD.
What I would probably do is to try and access the phone through ADB (if possible) repair the phone/screen, copy local files over, deactivate google drive backup, and just restore from local backup.