No such assumption was necessary for the original Copenhagen interpretation. The wave function cannot be observed directly, but possibilities for it are inferred from previous observations. Those functions evolve according to the Schroedinger equation, and can be used to predict observables. Making another observation allows us to restrict the set of possibilities for that wave function.
Physicists seem to be split on whether measurement causes a wavefunction collapse. If it does, then they are split over whether the collapse is objective, or related to observer consciousness. If there is no collapse, then they are split over whether part of the wave function escapes to another world where parallel observers see different outcomes.
I say that these dilemmas are all nonsense, and that the core mistake is to assume that the universe has a faithful mathematical representation and that the wave function is real. It is better to assume that there is no such representation, that there are no hidden variables, that there the wave function is not real but just a reflection of our knowledge about previous observations.
Matt Leifer gives a modern explanation of why hidden variables don't work. He has his own peculiar terminology, says that these are the only acceptable views:
Many people who advocate a psi-epistemic view also adopt an anti-realist or neo-Copenhagen point of view on quantum theory in which the quantum state does not represent knowledge about some underlying reality, but rather it only represents knowledge about the consequences of measurements that we might make on the system. ...So he says that maybe the wave function is real (psi-ontic) and there are some nonlocal hidden variables, or the wave function is subjective (psi-epistemic) and there is no underlying reality, as Bohr said:
As emphasized by Harrigan and Spekkens, a variant of the EPR argument favoured by Einstein shows that any psi-ontic hidden variable theory must be nonlocal.
There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature…Leifer cannot accept that, and insists on ontic states (where some math formula faithfully matches reality).
It seems to me that searching for psi-ontic states or for an underlying mathematical reality is just continuing to fall for the hidden variable fallacy. It is better to do quantum mechanics without any hidden variable assumption. Go ahead and assume that there is an underlying physical reality, but do not assume that it is faithfully represented by hidden variables. Observations have mathematical values, but not the physical reality.