You Americans are all so sue-happy
Opening public streets to racing doesn't work or it would have been done already. I think the main reason is because racing is a heavily regulated sport requiring all kinds of safety standards. You'd have to classify it as something apart from "racing" or it just wouldn't get approved.
I think the best idea, as mentioned, is to have public dragways which are heavily gov't subsidised whereby a small fee is charged. Kinda like the idea of phone booths everywhere would be nice.
But at the end of the day, it's just simply economics, supply and demand. There is not enough demand for these kind of facilities for anyone to build more dragways and hence the prices to participate will not go down.
As with the legal liability thing, not such a big issue. There is such thing as contributory negligence and assumed risk. If a boxer gets killed in a clean fight - that is, no malice involved - his family can't sue for wrongful death. If you participate in a dangeorus sport, you claim responsibility for anything that happens - as long as it is accepted as a possible occurence in the normal running of that activity. A simple waiver signing will accomplish that problem. So in my opinion, it is a tiny hurdle.