Stadia should have apps like YouTube/Hulu that open just like normal chromecast apps but designed to work with the controller. This would make Chromecast more competitive with Roku and other streaming platforms. Even barebones play/pause with the controller is better than nothing but a fully controller compatible chromecast app system would improve Stadia and Chromecast as a platform.
Have you tried using youtube and hulu through their respective apps on your phone? Now that you have a chromecast set up as long as your phone and the CCU are on the same wifi you'll see a cast button now. I'm 100% sure Youtube supports casting and I would imagine Hulu does as well(haven't used it in years). You can also use the google assistant (including the one on the stadia I imagine when it's enabled) to launch them on your TV.
I use Chromecast all the time. When I have the controller out and connected already, it would be nice not to need to pull out my phone.