The one reason we might think Jaresh-Inyo was elected by a public vote is that later on, he expresses the feeling that he is content to "represent his people" (supposedly, the Grazerite culture, although plausibly it might be a political party unrelated to cultures or species, too). Perhaps he feels indebted to his voters? But this is never directly stated.
Star Trek technology would certainly allow people to vote rapidly and comprehensively on a variety of issues, from grand to trivial. Say, the decision to essentially commit an act of war in "Redemption" (the blockade of the Klingon border against Romulan intrusions) might have been asked from the UFP Council by Starfleet, but ultimately voted upon by the UFP citizenry, in real time via an internet-like system. I rather doubt this is the case,t hough.