IMHO this is the fundamental mistake people make: conflating "I measured spin up" with "spin is actually up". Think about it: what is the evidence that there is actually a correspondence between a measurement and some external reality? There is only one possible answer to that: you can correlate this measurement with the outcomes of other measurements. For example, if you measure the same particle in the same basis twice in a row you will get the same result.
But there is a problem with that: if you actually look at the details you will find that it is actually not possible to measure the same particle in the same basis twice in a row because actually making a measurement requires entangling the particle with a macroscopic system, and you can only do that once. The best you can do is run a particle through a series of filters and look at where it ended up. You can then retrodict that the particle went one way or went another way, but you cannot actually measure its trajectory. In fact, you cannot even know that there even is a particle in your apparatus until the end when you actually measure it.
So the only thing you are left with at the end is a single actual measurement, and a macroscopic arrangement of some experimental setup about which you can tell a consistent story retroactively, but where you do not and cannot possibly have any direct evidence that your story is actually true, i.e. that the particle "actually" was spin up. And this is exactly what QM predicts: the consistent outcomes of measurements is not the result of the particle "actually" being spin up or spin down, but because the macroscopic systems (like humans) which compare the outcomes of their measurements are mutually entangled with each other and so decohere into classically consistent states. So if I ask you what you see as the outcome of the experiment, QM predicts that my perception of your answer will match my perception of the outcome. But it emphatically does NOT say that this outcome is actually real.