Every good CRT had a demagnetizing coil, that operated through a PTC resistor from the AC input. Full magnetization when the resistor was cold, then diminishing as the series resistor warmed. Bench demagnetizing coils did a similar AC/decay with manual intervention: you'd power up the coil, put your tool into it, and pull it slowly out of the bore and away from the coil. With regular use, your tweezers could drop a small ball bearing, it was THAT effective.
It scrambles the magnetic domains, but doesn't (at microscopic level) demagnetize so much as just make a lot of self-shielding opposing magnetized pairs.
Good luck with the toroid, but unless you know how it's wound, the windings mightn't do the demagnetization over the whole ring. An external coil will not have the poloidal-current-direction geometry you want,. Scrambling the domains is kinda hard when they're already stacked end-to-end in a circle.
Drilling ferrite is tricky: beware, internal cracks can turn it from a bell to a snare drum (generate harmonic, and non-harmonic hash). I've seen it happen.