I doubt it. Ever try to caption a video? It's a slow, annoying process. The automated stuff generally doesn't work that well so you have to carefully go through and fix errors and it's a giant pain in the ass. You then have to watch the entire thing to make sure that the caption timing is correct and that you've made it clear who is speaking when. For extra credit, try and make sure captions don't cover important parts of the video.
The problem with crowd sourcing is that you'd have to give a reason for people to bother doing it. The people who even can do it by definition don't need it. It's slow, it's boring, and it's annoying.
There's a reason it's so expensive to do, and that the government is forcing people to do it. Without government coercion, no one would bother.