Do you feel the shake in the steering wheel, or in the seat?
Steering wheel = Probably the front end
Seat = Probably somewhere else
When my Trailblazer had a shake that would make the rear view mirror useless, starting at 68 mph, worst at 73 mph, and gone by 80 mph, it was the rear U-joint of the rear drive shaft. Twice. The second time, I replaced the driveshaft. That Trailblazer now has nearly 270K miles on it, and I think it's just taken-out a front wheel bearing/hub assembly. I mostly hear the moaning/howling; but there's some shake in the steering wheel, too.
Even if the tires are balanced, brake drums or brake rotors might not be. As the drums/rotors rust away, the balance changes. Excess brake rotor runout can cause a shake even when the brakes are not applied...although it's usually worse when the brakes are applied lightly.
Skidding the tires wouldn't change anything in the "front end" that wasn't already worn-out. Flat-spotting the tires could cause a shake. But if he did this in his driveway, there probably wasn't enough speed to make a long-enough skid to flat-spot the tires.
Variations in tire sidewall stiffness is another possibility; but that would be consistent from the time the tires were installed. Also tends to be worst on low-profile "rubber band" tires rather than the taller-sidewall tires like the trucks use.