OB1
Jedi Master
I think it could be explained why the Dwarves several millennia later started digging again via the same line describing how the One Ring was forgotten about. To turn a phrase, "And some things that should not have been forgotten were lost. History became legend. Legend became myth. And for two and a half thousand years, the Balrog passed out of all knowledge."That's a possibility for this show. It really wouldn't make sense for them to delve too deeply and awaken something like a Balrog if they knew it was there, but it's a better idea than any other that I've seen so far.
While the second age timeline is being compressed, I don't think they're going to retcon the thousands of years that the film established passed between Sauron's defeat and the ring being discovered again, so it seems likely that the dwarves ended up forgetting (or ignoring) the old tale and re-awakened it.
Again, I have no idea how this all ties in with Tolkien's intent or writings, but it would make sense in the world of the film/show.