The tool shown is the perfect one, and surely someone near you who reads this has one.
ANY of the methods involving a split tool will do and be superior to thread files, because your threads are almost certainly compressed and not just surface damaged. A split tool, especially if started loose then progressively tightened, will try to roll the damaged threads back into place rather than just cutting across them. That metal is surprisingly soft...
Split a normal die, split a nut, find one of us that has the axle tool. For a little more money, "Little Giant" makes split dies that come with a special handle that allows you to progressively tighten the spacing until you reach the mark indicating normal thread diameter. If you buy it in 5/8--18 you will find multiple uses...every really troublesome part on a Model A uses that size!
Correct pulling should be done with:
A. Nut run to top of threads and left there till puller pops...
B. A small bolt, preferably with a dimple drilled into center of head, dropped into hollow shaft top as the bearing point for puller ram.
If you want a permanent tool, get a repro "bingo" wheel puller for $5 or so from any Ford vendor and cut it down into a cap for the threads.