QM is often called counterintuitive, but by itself, that doesn't mean much; many things get called counterintuitive, for diverse reasons.
It seems the real question here is, how often does QM get called "counterintuitive", with an implication that there's something about it which you just shouldn't try to understand. I say something, because obviously this attitude doesn't extend to everything about the subject. The physics students who Karl encounters on his campus are learning a lot about QM; it's just that they are also learning to not ask certain questions.
This attitude - don't ask why QM works - is held by many physicists, and endorsed in some teaching materials. But if the objective here is not just to ask a survey question about how common a certain opinion is - if the objective is to diagnose a problem in the culture of physics, perhaps with the further intent to identify a cure - then you shouldn't focus just on this attitude.
For example, sometimes the label "counterintuitive" is employed, not to block explanation, but as an explanation - a sort of meta-explanation. Why is QM so weird? someone asks. It's not weird, comes the reply; it just seems weird because your brain evolved to deal with the macro-world, not the micro-world.
Far more pernicious than this, are genuinely illogical pseudo-explanations of QM which pose as explanations. An example for which Karl has previously expressed a liking, is the Deutsch-Wallace decision-theoretic "derivation" of the Born rule for probabilities in QM.
Decision theory is about how to make rational decisions. Whether a specific choice has a good or a bad outcome depends on unknown features of the world, and rational decision-making involves considering both the probability of these unknowns, and the degree of goodness or badness of each unknown. So probability is an input for rational decision-making.
But Deutsch and Wallace construct arguments going in the other direction: they start with some "quantum game", and an apriori idea of what would be a rational way to play the game (in order to maximize expected winnings), and then they deduce what the probabilities would have to be, in order to justify the strategy they have decided on in advance!
Somehow I seriously doubt that the behavior of molecules in a gas cloud on the other side of the galaxy, is determined by the need to ensure that a particular strategy wins at quantum poker. If ever something were true, but still deserved to be described as counterintuitive, that would be it. But so far as I can see, it's not true, it has nothing to do with the truth, and the Deutsch-Wallace theory is just nonsensical sophistry.
As amusing as that example is, it still has basically zero influence among actual physicists. So at last we must turn to the mother of all quantum muddles, the historical root of the turn away from "realism" in physics, the Copenhagen interpretation itself.
Let me first dismiss a bastardized but extremely common version of the Copenhagen interpretation that has very wide currency, and which results from the natural reassertion of a realist outlook. This is the one which says that wavefunctions are physical objects, and they evolve according to the Schrodinger equation while unobserved, but when observed they jump to an eigenstate of the measured observable, with a probability given by the Born rule.
That is not how the original Copenhagen interpretation works. According to the original Copenhagen interpretation, wavefunctions are not physical objects, they are mathematical bookkeeping devices akin to probability distributions. What is real are observables that have been observed - a particle having an observed position or an observed momentum, a field having a measured strength at a particular place and time. Observables are what is ontological, wavefunctions are just a way to make predictions about observables.
I would regard this as the ideal philosophy to employ in teaching and using quantum mechanics, if we added one more principle: that quantum mechanics is an incomplete description of reality. Like classical mechanics, like non-quantum relativistic mechanics, it is a limited truth. There's something more. This is the sensible conclusion, and it's what many physicists believe.
What seems to have happened, at the founding of quantum mechanics, is that successfully applied positivist principles were wrongly turned into a new type of dogma. The difficulty in explaining the new observations with classical physics, eventually led to a deliberate abandonment of previous assumptions about space, time, and causality, in favor of focusing on that which was known to be real - the observable quantities.
Amazingly, a new, probabilistic type of mechanics was found which explained the patterns in the observables, without even positing, in the classical fashion, a definite state of the world between observations. But rather than say, of course the world is still there between observations, we just don't know the laws for what it is doing, it was decreed that the quantum laws are the ultimate truth; and we shouldn't worry too much about the lack of a "picture" of what happens between one measurement and the next, because measurements are all that really matters, anything else isn't physics.
Whether this attitude was good or bad in its effects is hard to say. Perhaps particle physics would have advanced more slowly, if more people had been trying to make this or that subquantum theory work. I suspect that the ultimate explanation of quantum mechanics is tied to recent ideas about the emergence of space-time in quantum gravity, in which case the right mathematics was just not available until recent decades. But whether or not that excuse is acceptable, I do consider Copenhagen anti-realism to be the root of many, many later "troubles" - other interpretations of QM which invent their own rationalizations for why we can do without objective reality, and would-be realist interpretations which settle for exceedingly vague descriptions of the basic reality. At least the Bohmians still know what it means for a realist theory to be mathematically exact.
Summary: Physics students and teachers who say "don't ask why" are just a symptom of a larger situation, and this attitude may even have pragmatic value for someone who wants to use quantum mechanics; but in that case I think it would be healthier to explicitly stick to the original Copenhagen interpretation, with an additional amendment about QM not being the final form of physics. Also, it is misleading to insert one's own favorite ideas about quantum or subquantum reality into the exposition of QM, in the benign guise of "making quantum mechanics intuitive". The essence of QM is the empirical predictive framework - that's why we can know it is incomplete - and anything more than that is a new theory.