The meaning of a result of any test of any kind is always going to be heavily mediated by the test itself. To clarify, I'm not talking specifically about the observer paradox, but rather the architecture of tests and experiments in general. As you alluded, the only competence a standardized test is capable of reporting is that of completing standardized tests.
While we can lament at length the pressure and dread of these events degrading performance, consider as well the situations, like the ratio of instruction of multiplication versus that of division, how the system can be gamed. Recalling my own childhood, I solved most of the answers on those tests through inference. In four candidate answers, there would be one which was blatantly wrong, two which were consistent with common errors and one correct answer. It was therefore often much quicker to deduce the answer to a math question than it was to compute it, and speed was important. What competence, precisely, does that demonstrate?
It might make sense to take a step back and look at the objective of the education system. What is it for? What is its mission? Is it to train an army of service labourers, or is it to prepare people for more preparation? Or, has its purpose somehow been lost along the way?
If the objective of the education system is to enable individuals to perform effectively in the contemporary world, it has a decidedly odd shape for its purpose. It looks a lot more like a daycare centre built to occupy and especially intern young people en masse while their parents are busy at work. The testing is more for the sake of onlookers, to assert that the entire exercise isn't a complete waste of everybody's time.
How, then, do we ensure that we are minting effective people? Can we even claim that's what we're doing? We can judge competence in two ways: evidence and testimony. The evidence we must observe first-hand. Testimony comes from a trusted source. Schools standardize evidence and manufacture testimony. Should they continue to be trusted?
The motivating narrative reduces to something like If you don't perform well here, you will never get into university, never get a good job and be doomed to a life of poverty and misery. PS: every burp and fart goes on your permanent record.
The ruse here is twofold: first, I wager my dental records are more informative than my transcripts, so the postscript ultimately amounts to an empty threat of Orwellian information disparity. Second and more importantly, the limited real estate and attention of trusted authorities to confer testimony entails that most students will not get to participate no matter how well they do.
When faced with lottery-sized odds, the most prudent strategy is not to play. If the objective is to become effective and prosperous, it may be worth examining the conditions known to lend themselves to such. Effectiveness depends on agency and mastery, which entails the space and time to concentrate our attention until we master a given skill and become confident that we can perform consistently. Prosperity depends on other people around us perceiving that our contributions are valuable. Increasingly, the latter is becoming harder to forge because of the growing resolution, resilience and availability of our collective memory. The easiest route to prosperity is indeed becoming the ability to be genuinely valuable to the people around us.
The question on my mind, then, is why wait to become valuable? What would happen, for instance, if children simply followed their curiosities, connecting with mentors around the world or in the same neighbourhood who had an interest in sharing their craft? There are many places in which even a young child would be happy to help out and be appreciated, and likewise many problems that could use the insight of a person who wasn't embarrassed to ask the wrong question.
Suppose our learning excursions, as well as our performances, were recorded onto a true permanent record, a sort of life portfolio which we could use to provide evidence, rather than somebody else's testimony, of our competence. Such an artifact would naturally need to be heavily curated, but that would be part of the process. After all, to initiate any social relationship is to pitch. It entails getting a person's attention in order to communicate why they would be interested in agreeing to your proposition. Creativity and innovation in this regard already happens within the current framework, but that isn't something schools teach.