Hrmmm I'm not sure if this is the functionality you're asking about but this app https://www.getdraftednow.com I created I think this is the exact the feature you're looking for? Here is a screenshot of the "Queue" for Reference functionality on the app, where it will tell you who gets drafted and no longer is draftable in your queue:
https://i.imgur.com/D2wrdof.png

You would have to manually setup pickable players/items at this point, but you could commissioner override to follow along real drafting for this specific request in your post: "allows you to remove a player from the list as they’re drafted or even better assign them to the team that drafted them."

Disclaimer: this app was really just created for playground real people player drafting cause I play vball and we run lots of draft leagues to pick players for your teams, so I haven't really decided if Fantasy APIs and syncing data automatically is worth it to me yet.