I have a Vizio Smartcast TV which is Vizio's name for Google cast or whatever Google calls it now. I am able to cast from Google Chrome to the the tv or from the Vizio app on my phone. I have found that when I am on Youtube and tell it to cast to the tv, it switches to that automatically. I also just cast a web page from the National Weather Service to the tv from Chrome. It swiched on its own. However, when I close the page, I have to use the remote to change the input back to DISH so I can watch the H3.
I think the switching happens automatically when you cast, but in my case I have to manually go back to the sat input. My Chrome browser which I only use for casting has a button that I got as an extension that allows me to cast any web page. It's a simple click. I believe that that button was made redundant in a Chrome update a few months ago. I still have it where I put it on the tool bar and it works, but I do not find it in my extensions list so I do not know what the name of it is. Also, if I ever uninstall Chrome or reload the OS on my laptop, I don't know if I can get that function back as I do not know any other way to cast to the tv unless the web page has cast function built in.
I do not think you can do cast with Edge or any other browser except Chrome. That's the only reason I installed it.