This is a very good point!

In fact, some people do argue that there is no measurement problem in the Copenhagen formulation of quantum mechanics to begin with – at least if you take it seriously and strictly go by the rule that the laws laid down by Bohr et al. only concern you as the observer and your knowledge about the system, and not the system itself. Following this train of thought, there is nothing "real" about the wavefunction and it is just a tool to come up with predictions. The same goes for the collapse of the wave function (which just describes a change in your ability to predict future measurements, and not a change of the object) and the term "measurement" (which we might as well replace with "enlightenment", i.e. the moment in which we obtain knowledge about the system).

In that sense, the only difference between classical and quantum mechanics is that our knowledge (viewed as a mathematical quantity) behaves differently in both theories: In classical physics, when we conduct multiple measurements of a given system in a row, our knowledge about that system will increase – to the point that, once we have measured all system properties to sufficient accuracy, we'll able to predict what any future measurement of any of those properties will yield (again, with some predictable uncertainty). So the knowledge of all our measurements has added up, it is an additive quantity.

In QM, this is fundamentally different: We can only know anything about the object the very moment we look at it. The rules of quantum mechanics (again, in the very strict interpretation laid out above) dictate that the second we conduct a measurement, we can forget about any knowledge obtained through previous measurements of other (conjugate) observables: Future measurements of those observables are inherently unpredictable. In that sense, our knowledge about quantum-mechanical objects never "adds up" to anything. (To see that this is really the the distinguishing feature between classical and quantum mechanics, recall that the existence of conjugate observables really is the only thing setting apart the quantum from the classical world: Without conjugate observables it would be impossible to distinguish, say, 100 electrons in a superposition of spin up and down from an ensemble of 100 electrons of which 50 are in a spin up state and the other 50 are in a spin down state.)

Of course, this whole interpretation is very unsatisfactory to lots of people (myself included) for a whole bunch of reasons. I assume that, to a large degree, this is due to the fact that laws of nature that put human observers in their very center seem rather undesirable. (At least since the time we switched from a geocentric to a heliocentric view of the world.)

But my impression is that there's another reason: Our intuition from classical mechanics & statistics has taught us that objects exist independently of us as observers and behave in a deterministic fashion, at least provided we as observers know enough about them. (Meaning that the more we know about the coin's initial position and velocity, the more likely we are to predict the outcome of the coin toss. If we don't know anything about the coin, though, the outcome is as unpredictable as measuring spin up/down in quantum mechanics.) Unfortunately, this whole line of argument is circular: The reason we believe that the existence of physical objects is independent of us, is precisely because knowledge in classical mechanics is an additive quantity and we can get to the point where we know "enough" to come up with deterministic predictions. That is, we never have to discard knowledge when running new measurements and so our knowledge takes on a independent "role" – which we call reality.