As someone who plays a relatively obscure MtG format, believe me when I say that I have vested interest.
If I were making a deck-list Extension, I would not try to leverage OAuth. Rather, I’d either add required configuration for the config view, or add options to live config view to let a broadcaster more easily change deck-lists on the fly, and not rely on a one-to-one mapping of deck-lists to broadcasters. Both of these methods require an Extension backend. I am assuming here that the deck-lists are publicly available.
If you wanted to avoid creating an Extension Backend, you could try to align Twitch ID with a unique ID in the system that manages deck-lists (this is only relevant if you control said system).
Deck overlay in VOD? Extensions and VODs currently do not intersect - an Extension overlay sits in an iframe on top of a video stream. You can always bake content into a VOD at the point where it is encoded. This removes the ability for a user to interact with it though.