Are there any problems that are known to be in a randomized complexity class (e.g. RNC, ZPP, RP, BPP, or even PP), but not in any lower non-randomized class (e.g. NC, P, NP), and whose membership in the randomized class is not based on the Schwartz-Zippel lemma?
If not, is there some fundamental barrier that prevents us from developing new tools? (apart from the obvious fact that we don't know whether randomization helps)