Or if you're talking about Amazon Prime Video, you just cast the tab because you can't cast the stream directly from the browser (like Netflix) or from any app on your phone because that only works if you're sending to a Fire Stick.

Somewhat tangential, if you have a TV running Android, you can install HBOGo but if you're on Comcast, they won't authorize the app so you can't use it. You have to install it on your phone and cast the mobile app to the built-in Cast receiver in AndroidTV (or do the same with the website on your other computer).

I know that Netflix was probably just an example, but for the record, Netflix is not one of these media companies.

You can directly cast Netflix when using it from a browser, you don't need to cast the tab.