the car tyres/wheels don't have pressure sensors, it's done off the ABS.
There is no reason why they couldn't put the new tyres on the front.
I was in Kwik Fit in February waiting for my M3 to be MOT'd.
A guy bought his Focus in 66 plate, said it had just had main dealer service, said it needed front brake pads, can u do it, here's the keys call me.
Five minutes after he left, drove car 5 metres onto ramp, without removing wheels, phoned customer and told him his discs were warped and needed replacing, extra £100. All up £180. Rip off !
So there's no way you can tell the discs are warped with 5 metres of driving.
Take everything they tell you with a pinch of salt.
I replaced the rear pads on my M3, had done 50,000miles , Automatic , so much harder on brakes, were half worn, say 4-5mm left. No lip. So I doubt yours are that far gone. Most of the braking is done by the front wheels.
My cars is on its 3rd sets of pads on the front, same discs.