A machinist turned me on to Tap Magic a couple of years ago, not sure how it would affect finishes but from a thread cutting perspective nothing beats it.
That's really a matter of opinion and the material being tapped. I don't particularly like Tap Magic and won't use it if sulfurized cutting oil is available. If you're tapping cast iron, use kerosene as a cutting oil.
Broken tap removal tools are hit and miss. Sometimes they work and sometimes they just break which compounds the problem. I've never had good luck with them and don't even bother trying them anymore.
Sometimes if the tap isn't stuck too badly you can use a hammer and punch to back it out of the hole. If that doesn't work, a carbide endmill will cut that dude into dust. There is also the option of EDM removal with either a hole popper drilling type machine or the possibly more common RAM type machine.
I've used a lot of methods and sometimes a combination of them to get broken taps out. It sorta just depends on the situation as to which one is best or gets tried first.
Machine tapping, if done correctly will reduce broken taps (not to mention crooked holes). Remember plug, taper and bottoming taps are not very suitable for machine tapping. A spiral point or a spiral flute tap is the best choice.
O rly? Spiral point taps push the chips ahead of the tool, which is fine for through holes, not so good for blind holes. Spiral flute taps are great for gummy and/or soft materials(aluminum for example), but don't work very well in hard alloys and are more prone to breakage since they're even weaker than straight flute taps. The other types work just fine with the proper cutting oil/coolant mix.
Thread milling for the win--100x better than tapping.
FWIW I don’t use taps that you could hit with a hammer and they would "fall apart", That would be asking for trouble from the start.
For some applications, a carbon steel tap is desirable. They will work just fine with the proper cutting oil and are easier to break up if one does snap in the hole. HSS taps can be a major PITA to get out in some situations.