Apple TV Paramount+ App Live TV not working

The app is not built in, it has to be downloaded from the App Store. However, you are correct that there is a different version built for the Apple TV platform.

If the team at Paramount Plus is unable to resolve the issue then it would be up to them to investigate further so they can send out an update to fix their app.

Unfortunately it is not uncommon for providers to pass the buck like that. Yes, if you billed it to your Apple account then the refund would also have to come through there. However it is still up to Paramount to fix the issue with their app. So it's up to you if you want to try contacting them again and escalating the issue so that this can be fixed.

I use the Apple TV Channel subscription for Paramount, which is done directly through the Apple TV app, and find it much better. We don't have the Live TV feature here in Canada, but you can check if they offer it in your location. Since it is considered a separate subscription you would be granted a trial if it were your first time using it, so no harm in trying to see if it works better than using Paramount's app.