By the way - what would be wrong with just using tap/die set to cut in new threads up one size if I screw the old ones up? Is that not a good idea?
It may work, but not the best idea.
Often there is not room for the next size bolt, and you'd also have to drill the hole in the other part that it goes through larger as well.
If it comes to that the heli-coil is the best option, nothing is changed other than the (better) quality of the threads, still use the same size bolt.
If you really mess it up bad there are threaded inserts that go from one size down to the next smaller.
Try welding it first,
Or if you're set on trying the extractor be sure to drill in the very center. If you're off center you risk damaging the threads, and if the extractor doesn't work the of center hole makes any other method harder.
Grigg