Not sure how many neoreactionaries actually agree with that definition. But anyway...
So the belief is that societies that (1) accept democracy (2) will inevitably (3) meet their progressivism-caused doom.
(1) We would need a working definition of "democracy". Specifically, what about countries like USSR or Burma or North Korea that nominally have elections, but the winner is reliably known in advance. Do they also count as "democracies" for the purpose of our belief; that is, does even half-assed pretext of democracy inevitably bring the doom? Or do we need people to participate in real elections? What if the elections are real, but most media are in hands of a few rich owners, and most voters believe the media?
I am asking this to avoid rationalizations from hindsight, like: "Singapore seems to be doing pretty well despite being a democracy -- nah, they are a democracy only in name, it's actually People's Action Party ruling since 1959"; "North Korea seems like hell -- well, they do have elections, so this is an example of a democracy that already met its doom".
(2) The word "inevitably" actually doesn't predict any specific outcome, because if the prophesied thing didn't happen, you can always add "...yet". Could it be made a bit more specific? For example, do countries with a lot of democracy meet their doom faster, on average, than countries with only little democracy? (For example, should we expect Switzerland to meet their doom sooner than North Korea?)
I am not a neoreactionary, so you'll have to find somebody else to argue their side -- shouldn't be a problem, since you mentioned that they can't shut up :-P
However I'll explain why I find their ideology interesting. The thing is, in contemporary political discourse in the West democracy became a sacred cow. One could talk about better or worse implementations, point out issues with specific governments or policies, etc. but the notion that democracy is the best and you should always try to have as much of it as possible seems to be sanctified, enshrined,...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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