You are correct, but modern receivers will actually use the transmitter like a "NDB" and will detect that the incoming position is not the same as the stated in those messages, ignoring them.

The same thing happens with sudden changes in GPS messages that are completely out of alignment with the inertial system, they will be ignored. (unless you somehow are capable of knowing exactly where the plane is and gradually hack the GPS messages)

ILS systems are maybe the most dangerous and prone to hacking, but keep in mind that those frequencies are under surveillance and police is dispatched within seconds of a clandestine transmission on those frequencies. Again, here the gps and inertial is also used to guarantee nothing completely wrong is accepted as valid ILS sources.