my browser, the way that I use it, does work only when connected to the Internet.
Offline web apps are awful for me. I don't live in a far away village and I'm never without access. Your use case is different, that's fine.
I'm not arguing can/can't or should/shouldn't. In fact, I'm not arguing at all. I'm pointing out that Chromecast is a connected device.
If there were demand, Google would add offline support. If there were demand, Roku would do it. Or Amazon. But as I already mentioned they can't make online Casting/Flinging/Flicking work reliably.