A tire balance has nothing to do with an alignment. Wheels and tires stay on for the alignment so they can drive it onto the rack and hook the target boards onto the wheel. An unbalanced tire will not impact caster, camber, or toe. They are separate jobs. Just like how an air filter change and oil change are separate. But I guess plenty of people believe service writers when they say it all needs to be done at the same time..

OP, you can request a printout of the alignment specs. If they say their printer is down, ask them for a simple pic of the readout screen that the tech uses when they put it on the rack. A lot of chain places like Mavis have a free alignment check, so it shouldn't cost you anything but a little more of your time. If everything is in the green, you're good to go. The only reason I can think of as to why your wheels were off would be if you have aftermarket wheels that aren't compatible with the target board clamps, and they happened to have some that would work on hand. The odds of that and the odds of them going through that effort are like 0.01%, though.