I just had a quick look at telgram and it may be fairly easy to get most functionality working.
What happens if you type the name of your telegram notification group/service in the notification group section? do you at least get the title and message? If there is an error, please provide it.
The blueprint relies on mqtt messages as the trigger.
The MQTT config in frigate.yaml is meant to contain the username and password of your mqtt application (mqtt addon in your case)
You can setup a login for frigate in the config for mqtt
I have image notifications to the tv working in beta 8c.
If you want to do a group that has both android TV and android phones you can notify both but you have to chose which one gets the snapshot using the TV Notifications toggle. Otherwise you can still continue to get snapshots on both by having two separate automations.