The broken tap problem lead to the Tapmatic:
Which is a wonderful tool if you are tapping many holes (but a bit pricey for the HSM). I have always tapped holes by hand using a guide or adapting a mill for the purpose - with plenty of cutting fluid and judging by hand force taps don't bread - but it is slow.
I think the job would decide how to remove the broken tap. Generally, work is done for profit and time is money so the quicker the tap is removed and threads formed the better. Machine shop forums are a better place to ask this question if you are looking for experience and ideas. YMMV.