@Joseph Just looked up the Nitei Gavriel after shacharis. You are correct (and you got about 95% of your comments and ma’arei mekomos from his extensive footnotes). He is very clear, though, that it’s almost exclusively a Chassidishe minhag and only in Chutz La’aretz. So there’s a good reason why most people today don’t hold of it.
Also, although he doesn’t say it outright, he seems to imply that there’s no known reason for the minhag, just a lot of possibilities brought down by various poskim, and it’s just מנהג ישראל תורה. That’s enough to not criticize Chassidim for not learning, but still a very shakey thing to have to rely on. So I guess that unless you’re from a Nusach Sefard family, don’t use a metal object to cut your beard and peyos, hold of exclusively Chassidishe shechita, and all the other Chassidishe chumros, there’s no real reason to be mevatel Torah on either night of Christmas.