I'm not knowledgeable about the superstitions but I wouldn't be surprised how far they can go. Technically, placentas are still "meat" and do contain a lot of hormones as it's an endocrine organ so the claim of it being nutritious isn't hard to imagine. But I haven't read any extensive studies to support their consumption on mainstream medicine.
The lotus birth is something I came across and only heard some celebrities from my country do but local lore wise, there's burying the placenta at home to ground the child to family or drying up the umbilical cord to be used as a bookmark to bless the child with intelligence.
Keeping the placenta attached as long as it can seems like inviting possible infection on the cord but I'm no expert on the subject to make a comment about it.