But what do you do when a large group of people completely reject the idea that experts are actually experts, instead substituting non-experts who will say whatever is popular rather than objective facts?

You let them decide anyway. Through whatever the political processes are, of course - we have representative democracy, not direct democracy or referenda, for the most part. But ultimately, in a democracy you let the people make these decisions. If people have ill-informed ideas about public policy, it will be nothing new. I don't believe the electorate has a much worse grasp of "the objective facts" that surround complicated issues of public policy or economics or foreign affairs than did the 90%+ of voters who hadn't graduated high school a hundred years ago. The legitimacy of a democracy isn't contingent on the "quality" of the voters.

Reading through some of the posts on this thread, can you not see why some voters were not entirely convinced that it was the Republicans that were a threat to democracy, rather than the Democrats? Suggesting that choices on major issues of public policy are deficient unless the voters meet some external standard isn't exactly a thrilling endorsement of democracy. It's actually more in line with retrograde limits on the franchise, when the Founders tried to make sure the voters were of sufficient "quality" by the expedient method of only letting the "right people" vote.