I am running the same firmware version on the same app version on Android 13 for Moto and experience near instantaneous load to live stream from both the Home Device List and when the cam is grouped. I cannot replicate what you are experiencing.
My only suggestions, which are more standard troubleshooting steps than possible solutions, would be to (1) power cycle the doorbell, or (2) uninstall and reinstall the app, or (3) uninstall and reinstall the doorbell. The last would be to contact Customer Support and log a Support Ticket.