You absolutely do NOT need to keep Netflix running on your phone and you absolutely CAN use your phone for something else while the Chromecast is streaming. I've been using a Chromecast since they came out and love it. In fact, my son is using it to watch cartoons on Netflix right now and I started the cartoon via my phone. I just rebooted my phone to see what would happen and sure enough, Netflix kept working just fine on the Chromecast. Once you start the stream, the Chromecast itself is what is communicating with Netflix/Youtube/whatever.
One thing you are correct about is the local media streaming. That is a weakness, but you're incorrect on just about everything else in your post.