they are nonetheless legitimately extremely popular among their citizens. Maybe they rigged the votes for insurance, but it seems entirely possible to me that they would get the support they needed either way.

I doubt this. They have some significant support for sure (as does Trump), but I don't believe that if you where to have an election with close to 100% voter turn out they would remain in power for long.

But existing democracies are already trending towards dictatorship, so it's not like you could do much worse.

That's my whole point. There isn't really any evidence to support this claim (for direct democracies). If anything it's the opposite. Representative democracies may tend toward dictatorships (power has a tendency to trickle upwards), but direct democracies with high voter turn out rarely do. (Unless you have some evidence otherwise?)

(And to be clear I do have problems with direct democracies too, but my feeling and the evidence I have seen points towards the opposite. Western democracies need to become more democratic, not less.)