As a fan of the show, I really enjoy this idea. Have you play tested it at all? Sounds tough as nails. I personally dont mind the extra pieces and dont find it that difficult to manage. in fact I would encourage you to expand the idea a bit more.
I would love to see a Castielle mechanic come into play. Maybe something to do with being able to teleport to locations in need.
Also where are the character cards? no Dean and Sam players? Maybe Castielle is another player character? What about Bobby? or you could go another route and make these guys allies? depends on how you implement it.
It seems like alot of work to go around getting all the rings and without the weather cards in the deck i fear the gates would open at an incredible rate making it even more tough. maybe a mechanic to slow that or more dooms on lucifers track are needed? its hard to say without playtesting.
The idea is great and I think it does fit the game theme nicely. I also like the ideas that the players have mini quests to do, like get the rings, instead of just running around sealing gates, but unfortunately theres nothing stopping them from ignoring all the new mechanics and going for a seal victory, other than tougher monsters.