Just following up as I completed this repair project. The tap came today and I put it right to use.
Once I ran it into what remained of the old screw, it pushed it right out.
I was a little concerned that maybe I had taken too big of a bite out of one side of the hole, as obviously the drill through wasn't perfectly centered. But after cleaning up I greased the new adjuster screw and installed it. It went in cleanly with good engagement. Now both dropouts sport fresh adjusters.