I sort of have the values of Medieval Catholicism too. My theory is that people who are neo-reactionaries as you define them, probably had past lives during that era. So they feel connected with it or resonate with it. I certainly feel like I have had past lives during the Middle Ages and Renaissance in Europe.
If you research past lives, you find there is a lot of compelling evidence for it that cannot be explained away, which has been a thorn in the side of Christians and Atheists, because their narrow paradigm and mental box doesn't allow for it. And in fact, reincarnation was part of Christianity until 553 when Emperor Justinian outlawed it. Dr. Ian Stevenson compiled hundreds of cases that cannot be explained away. Are you aware of all this? Recently some cases of reincarnation went viral and was featured on the news too, such as James Linberg, Jenny Cockell, etc. Did you see their stories? They are mind blowing and have convinced many that reincarnation is real. So you can't dismiss it just because it doesn't fit into your belief system.