You are right to be afraid of EZ outs. Some basic rules of using them are..
If the bolt breaks removing it due to corrosion forget the EZ out using only a small hole in the middle of the bolt, chances are it will break off, this is especially true in aluminum.
Not getting the EZ out deep enough in the broken bolt, if it is only in there a few mm it is a guaranteed break.
Spiral types are the most common and usually cheapest, these are the worst for breaking, they can actually jam the broke piece tighter in the hole. The tapered type is better, only use high quality (more money).
EZ outs can be used successfully on bolts that broke going in, this is common when TTY bolts are reused too many times, using a LH drill bit will get them out most of the time without using the EZ out.
They are also successful when most of the bolt is gone and only a thin portion and the threads themselves remain eg a 13mm bolt drilled out with a 11 or 12mm bit, the remnant will come out easily.
Accurate drilling is essential for easy removal, basically you must make the broken part flat if proud and use a center punch that is exactly in the middle of the broke part and drill straight. You may need to make a jig out of 1/8" plate to attach to surrounding bolts with a bolt tapped into it. Drill through the bolt on a drill press so the drill continues going straight as its going into the broken off piece.
Your broken part should be very easy, it already has a hole going right through, just use the next size up and continue till its almost gone but the threads remain, what remains will come out easily with no rethreading, just chase the junk out of the threads with a tap (if its pipe thread you must use a pipe tap) in that case if the threads look okay I wouldn't bother, if you start cutting new threads it will sink the part deeper and cause problems as pipe taps are slightly tapered.
You can use a Dremel with a small wire cone brush to clean any rust off the threads.
I have had one of these for many years, its the best center punch I have used, it takes a little practice but is more accurate than automatic punches and easier than a hammer and punch.
https://www.amazon.com/Spring-Tools...mp;psc=1&refRID=XP5Z1FM1VQ3GDM7J78MA