Is there a non-welding option?
There are some, but there is no comparison to the welding option.
The same method I use with the TIG can be done with a MIG welder, although not as easy on the hard to get to ones, or screws broken off below flush.
Even a stick welder will work if you're good enough, or if the bolt is large enough.. I've tried them all on broken bolts and screws, and prefer the TIG welder.
There are some pretty affordable MIG welders available, even small ones that run on 110V would help remove screws.
Grigg