Screws typically break off to leave an uneven surface. Its very difficult to get a drill to start just where you want it on the end of a broken off screw.
The true running rigid spindle ,precise x and y positioning and rigid vise make a mill the ideal tool for the job.
But a drill press can do the job.
Do very carefully position/secure the workpiece in a vise under the spindle.
Now what you need is a small center drill. What might be called a "combined drill countersink" in the tool catalogue.
The original purpose is to drill those nice center holes you often see in the end of a shaft. The lathe tailstock center rides in those. Tts common practice to use a center drill to locate a precise starting point for a twist drill.
Use a center drill to create a "start" for your drill. They are short and rigid and they don't wander off. Get the center drill pilot started then some of the 60 deg countersink. Now you can start your drill.
One trick that sometimes works is a left handed drill. Its ground to cut in the counterclockwise direction. Generally as a drill breaks through it will catch and grab on the workpiece, and the stub of the broken screw will be heated b the friction of drilling. The left hand drill generally backs the screw right out.
Some EZ outs look like a left handed licorice twist. Some are just a tapered square with a groove for some "hook" at the corners. I've had the most success with the square type. I grind my own out of a lathe tool bit and use a tap wrench.
I'll throw in a bonus trick. Suppose you are changing the car's thermostat and break off a bolt. Pretty hard to start a hand drill in the center of a broken off bolt under the hood.
But if you look at it just right,that thermostat housing will do for a drill jig.
You just need a bushing that will just fit in the hole the broken bolt goes through. Its OK if it fits with a tap of a hammer. The hole through the bushing needs to just barely let the drill pass through. You can drill the hole with the drill bit size you are going to use to do the job.
Now,use the unbroke bolt to hold the thermostat housing in place, eyeball center it the best you can, tighten the bolt and at least you are hand drilling with a drill jig and drill bushing. It will probably work.
Beware a cast aluminum part will often be fitted with a helicoil or Threadzert
I'm not sure how that will affect the job, but keep your eyes open.
You may end up replacing a helicoil
