From my personal experience, that is ballocks!
I was in the same situation, bought car as an Approved Audi but only the two full function keys - in my case Infotainment screen showed that as you would expect 3 keys had been paired to that car, so I bought a new plastic key from Audi and handed them the 2 full function keys and the plastic one, job done, all 3 keys work and infotainment system shows 3 keys paired to my car.
If they are saying that the plastic key can not be coded, you could use it not only to open some other cars, but to drive off with them, Audi workshops need a bit of training I think!