In the frum community, the argument is referred to as "the Kuzari argument." I don't know the name of the fallacy, but please do lmk if you find out.
I read that .pdf multiple times 5-7 years ago, and the Kuzari argument assumes that religious people assess dogmas the way a scientist would assess a scientific claim, which, like you said, is fallacious.
I’ve recently been reading Rabbi Dr. Gottlieb’s book “Reason To Believe,” which promises to give “sufficient evidence” that Judaism is true (and in my personal opinion, has failed rather spectacularly as of what I’ve read so far). One of the central arguments given is that Judaism contains a legend that could not have been falsified (and if that legend is true, the rest must follow). The reason for that is because the legend is supposedly one-of-a-kind, (specifically “a national event of a type that would be hard to forget”—I believe that any proposed counter-examples given would be dismissed as not having enough witnesses present, while this case supposedly had 600,000 witnesses), and that the problem for the critic is that there are no other known cases anything like it (supposedly) to draw evidence from. I don’t even know where to start with this fallacy, I don’t even know what to call it by, and would appreciate it if anyone could recommend further reading on what exactly is going on here, logically.