You keep bringing up MS Cloud Gaming, but there are others that are more established.

https://www.nvidia.com/en-us/geforce-now/

https://luna.amazon.com/

Luna can only support iOS by using Safari, for example.

Does Apple have a rule that says beta apps aren't allowed on the app store?

As far as I'm concerned Microsoft cloud gaming is like a 1.0 version and works fine on Windows and Android. I had no idea it was a beta product until just now.