Why is it not possible to build a competitor to Chromecast? Sure, you'd need to manufacture and sell a hardware dongle*, but that's not impossible. On the screencasting front, it's possible to screen-grab from Android and send that so you could mimic built-in functionality (not sure about iOS, but if Chromecast can do it then other apps can do it). Casting from media apps would require tie-ins with media companies, which again is tricky, but they're generally a bit pissed with Google so would likely welcome a competitor to the space. You'd have no chance of getting Youtube on board, but then you've got a decent case for anti-competitive behaviour, which EU and US systems are circling Google for anyway.

I'm not saying it would be easy, but it's certainly not impossible, with the right business model and business and political environment.

* It might be possible as a smart-TV app, though smart TVs tend to suck which is why Chromecast still sells units