Your logic is valid but a careful look will tell what is happening. This strange behaviour is not that uncommon, it has analogy in other areas also. For example when you push a large stone, it is pushing you back (famous third law), then how does it move at all, if it cancels the supply force? On the other hand when you push a strong wall, it reacts with equal and opposite reaction. The thing is if the wall did not exert this reaction it would have allowed itself to be pushed back without any resistance!! These action / reaction like, supply voltage and induced emf (other examples are terminal volt of a d c motor and back emf, generator output and its reaction on prime mover etc etc) are all must be conceptually understood. I can dare say that they do not exist as such! They are merely way of understanding the concepts. When you supply ac volt to a coil (transformer without secondary load is just a coil) because of energy transformations taking place we see that the coil offers resistance much different from pure resistance, which is called impedance. But the same can be viewed as reduction of available voltage due to induced emf. I repeat this is a conceptual understanding of the phenomenon.
Leaving aside the double negative, the problem is your use of cancel - what do you expect - that it will go back to the generator and kill it? The transformer produces an output voltage which tries to do work against a resistance or inductance - the work it does produces a back EMF which the supply voltage works against so the generator has to do work. If there were no EMF in the primary, it would behave like a short circuit and blow the circuit - which is does in a small scale in the starting surge that you have to allow for when turning on equipment.
