It's a PITA for sure, EDM is ideal solution but generally out of the reach of backyard or home workshop types.
A former colleague always had reasonable success by tapping around radially with a punch if anything was sticking out, but the punch usually gets pretty chewed up.
Another guy that worked with me showed me a trick when we had a tap break off in a mold that was in for a rush job,
We put pins in between the flutes of the tap and kept wiggling it around until it came out, but it required a lot of expletives and angst to finally get it out. IIRC, it may have been old-school bobby pins, but for sure the tap wasn't jammed too bad, but nothing was sticking out either.
Also saw a guy weld a nut onto the protruding piece of broken stud or tap, combination of heat cycle and better leverage worked ok on my brothers diesel rig.
No one size fits all solution though for sure.
Live while you're alive.