The point is that in some interpretations, like dBB, the measurement process is covered by the theory. And the essential point of preparing the wave function of a system is to put the trajectory of the measurement device into the wave function of system + device. (In this sense, the argument is indeed interpretation-dependent - which is natural, because I don't know how to make arguments which would make sense in interpretations I reject as nonsensical, like many worlds.)Sorry - but I don't get that at all eg why even the concept of 'prepared' is applicable to wave-function needs detailing before you can even introduce it in that context. In modern times state and preparation procedure are pretty much synonymous but that requires detailing of what it is in the first place - which is very interpretation dependant.
But in this case, if you want to apply this to the preparation itself, and not just to a measurement, which, then, leads to a subsequent preparation of the system after the measurment, you have to start with some assumptions about the unprepared wave function.
Fortunately, this is not as problematic as it looks like, because in this case you can use epistemic considerations to define the initial wave function. You know nothing, thus, can try to find out which wave function or state accurately describes this knowledge.
The point of the argument is that after the preparation measurement, the effective wave function depends on the trajectory of the measurement device, which is something really existing even in the Copenhagen interpretation.