Can someone please explain to me about the whole "finishing things up" issue?

Definitely a strange issue.

Could there be a chance that this console is bugged, modded or broken preventing the downloads from working properly.

I tried this same game, same DLCs on the same Xbox Live account, across 3 different consoles in three different locations, yet only happens on this console, even copying from a console were the DLCs were working doesn't seem to work either.

I see. May I know did you install the game directly to the internal storage?

Oh, that's a good question. There's no option or way to download it from there though. Rest assured reaching the game developer will do the trick.

I seem to only be experiencing this on this specific game.

Other games work just fine

Just to clarify, is it happening on your other DLCs?