Kill Your TV. Go outside and play! :) This is the best advice yet on how to deal with Chromecast. It is also therapeutic and prevents zombieism. There is truth in that. On the other hand, I have yet to hear anyone say, "Kill Your Library," since libraries are also filled with garbage.
Chromecast and public libraries are tools. How we use them is what is most important. Chromecast and cable are filled with jewels and garbage. But Chromecast also offers greater control, and greater jewels.
There's no need to throw the jewels out with the garbage, unless a person cannot restrain himself from watching garbage.
I would not go so far to "kill my library" but there is a difference between being passively entertained versus active participation (reading, for example). A study came out recently about the deleterious effects of television entertainment (don't have a link, sorry) but your point is well taken concerning garbage and jewels. Most of the classics I find to read come from the Internet. Most of the junk I pass up comes from the Internet.