Since I was going in a Cincinnati direction with the chili spices--cinnamon, cocoa, coffee--and those are flavors that beets can work with (have you ever had beet chocolate cake? It's pretty good.), I decided what the heck and tossed it in. It's not bad. Really, it's too mild to make much of an impact against all those other strong flavors, but it adds a little sweetness and a little tartness missing from the mix otherwise. Also, the beets do make a contribution texturally. I used beef shanks so I have shreds of meat instead of chunks; the beet bits are the only component with any bite to them.
Upon consideration, I've decided that I should have pickled the beets first. Then I'd really have something.