A very long time ago, I went to university and took CS. I found it too easy, so I switched to Physics. I found that too hard, so I switched back to CS ;-). I didn't get "the lecture". In fact, just the opposite. We were taught physics and then told that it was up to us to learn whatever math we needed to help us. Not being particularly fluent in math, I just couldn't keep up at the time. One of the best things about my brief foray into Physics was that it taught me the importance of learning math -- not as a prerequisite for something, but as a tool.

Having said that, I think you are wrong in your supposition. QM is hard to reason about because it doesn't manifest itself particularly visibly in our everyday lives. The framework that you have to work with comes from your own imagination (and mathematical models). There is no real problem with that, except that if you get the axioms wrong, you will necessarily get the conclusions wrong as well. There are an infinite number of ways that it could work and virtually all of them are incorrect.

With classical physics, you can reason based on your lifetime of observations of everyday objects. Of course, we still get it wrong. The more we abstract away from the ordinary, the easier it is to jump to strange conclusions. QM is abstracted so far away that it is incredibly hard to test anything intuitively. That's why math is attractive. It gives you a logical system for testing your ideas.

Math is just a language, though. It's a way of expressing an idea. If you have an idea and express it in English, it is no better or worse than expressing it in Math, and no better or worse than expressing it in pictures (Expressing it as short presentation with hand puppets, is obviously the best way, but that should go without saying).

Similarly, if you are working through a problem by talking to yourself, or doing math, or drawing diagrams, or imagining scenarios with hand puppets -- it's all equivalent. The representation of the thing is not the thing.

And therein lies the problem. Many people do not really understand things well. They think they do because they can picture a representation of it. Let me give you an example. I once showed a colleague one of those ring-within-a-ring magic puzzles where you have to unlink the rings. I showed him how to do it. I got him to do it 4 or 5 times and then I asked him, "Do you understand how this works now"? He said, "Yes, it is very simple". Then I took the magic rings away and gave him some straight wires. "Please build the magic rings", I said. Of course, he couldn't because he didn't understand them.

We see this problem all the time in programming. People think they understand the problem because they can write code. The code that they write gives them an illusion that they understand the problem. But the code is not the problem. They start focusing on the code and forgetting the problem. Eventually they have written a huge amount of code and they have not solved the problem.

You're just noticing the same phenomenon in Physics. It doesn't matter if a person builds a model with math, or diagrams, or English description, or hand puppets -- most people will start to focus on their language and their tools and become dissociated from the problem. They create their own religion around the model into which they have poured their creative energy. That becomes their reality and they have no idea that it doesn't match a more objective reality. The more abstract the system, the easier it is to tumble into fantasy land (which is why programmers are so susceptible IMHO).

The lecture I wish people would get is: "You are wrong. Now tell me why. If you don't know why, find out." That would help people a lot better, I think ;-)