<blockquote>I've often thought that it would be great to crowd-source funds for e.g. a TV show or a movie, but it really is very difficult to raise that kind of money on the promise that the donators will like the result. Medical research is the same kind of problem (only more expensive).</blockquote>
Showtime, HBO and others are examples of what you seek. They survive on subscriber funds, they produce new and innovative entertainment and IMO the best shows on television now. They do so without commercials (and the interference they bring) and without (AFAIK) government intervention or assistance. So you shouldn't need to wonder as it already exists, but costs considerably more than commercial television.