problem with veiwing apple tv on my smart tv

I have the following devices:

iPhone 13 Pro

MacBookPro (OS 10.11.6)

Sony XBR55

Chromecast (older)

I just signed up for Apple TV, and I can't seem to find the magic formula to either cast from my iPhone or watch on my Sony smart tv. Apple TV doesn't show up as the pre-installed apps on the tv. When I do a search for Apple TV on the tv, it says "your device isn't compatible with this version."

So, what do I do to watch Apple TV on my smart tv? Is there a way to cast it from my iPhone? Do I have to download something?

Thanks!