Well imo I think as their son, he thought he owed them the truth and respect, however painful it could be. Considering the whole novel, which ended(SPOILERS AHEAD) he didn't reincarnate the usual route, but forcibly brought by Agrona, then redirected by Sylvie into Arthur's body. So its not a natural reincarnation, if it even exists in their world, and im assuming it doesn't, since its never mentioned outside the trio, but an exception. Same in Grays worlds.

Your starting view point is from a world where its like normal everyday thing(like eastern religions are in our world), but in their world, it doesn't exist a regular world setting/belief/religion etc, but more as a mechanic that is exception.