This always bothered me but has anyone else ever found it weird that throughout all it's inceptions, Hawkman (more specifically Carter Hall) has never been incorporated onto any iteration of Justice League Dark?

I mean we're talking about a guy who's origin and most defining characteristic is that he's locked in a (seemingly) endless cycle of reincarnation, and given that many of his earlier escapades dealt with him looking into his past lives one would think that Hawkman's lore should be steeped in the occult. And yet despite that DC doesn't really seem all that interested in using Carter in any way beyond the JSA or regulating him into being just another alien. Does no one see the potential of having Carter on a team of mystical characters when he could essentially just pull knowledge from one of his past lives and is running around using weapons more or less designed to deal with magical threats? Why would you not want someone like that on your team? Hawkman is literally the closest thing to the Avatar that DC has running around, and that aside is still a very capable fighter and archaeologist capable of bringing a lot to the team.