If the car didn't shake at 60 with the old tires, it shouldn't shake at 60 with the new tires.
If a bent wheel were the problem it would have caused the same shake before the new tires were installed. You didn't say there was a shaking problem prior to the new tires.
"Only one true balance for a car." What the heck does that mean? Whoever said that is nuts, and not to be trusted.
It will not get better with tread wear. If you keep driving with this imbalance your new tires will wear prematurely, which will make things WORSE, not better.
Normally I would suggest you go back the the original installer and have the tires re-balanced, but in this case I'm going to suggest you go to another shop, perhaps one specializing in tires, wheel balance, and alignment, and let them check the wheels, the balance, and the alignment.
And then I'd also suggest you find a new mechanic. The one you're currently working with is incompetent.