People are rarely so forward thinking as to fund research that isn't likely to be complete before they succumb to their own illness, much less to do so proactively on the chance that they will develop a given disease.
"Why should I fund cancer research? I don't have cancer!" -- young, cancer-free person.
I'm not saying that it couldn't work, I'm just doubting whether voluntary funding like this would be sufficient to raise the necessary cash. That's the advantage of government funding; governments are more suited to long term investments and rewards.
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).