Sorry, I don't know why I thought the centering cone was threaded. Complete mistake on my part. It's not threaded.

I managed to align the centering cone so that it bears down on all the bearings evenly by moving the fork around as I tightened down the top cap on the stem.

The way the centering cone is angled toward me, along with the fact that the top cap was very loose when I first started unscrewing it, suggests to me that my tugging at the handlebars in the past week had loosened the top cap bolt and caused the centering cone to go out of alignment.

Good thing I found this now, otherwise I might have been looking at some new bearings at the very least!