The Broken shell extractor has been part of every armorer's kit since 45-70 days, if not earlier. Usually in the machine gun tool set, too.
They are caliber specific, but not cartridge specific.
.22 won't work in a .30 bore, and .30 won't FIT in a .22 bore, etc.
.30 cal will pull ruptures from .308, .30-06, .303Brit, 7.62 (39 & 54R) MIGHT work on 8mm mauser.
You CAN use a coarse thread screw or a tap, and cut into the fired case, then drive it out, BUT, if you cut through the brass, you're cutting your chamber wall, and that's generally not a good thing.
Since you're not in a firefight, and don't need to clear the MG before the next human wave attack, its foolish to do anything but take the time to get one of the extractors and use it, or take the gun to a professional smith.