I majored in mathematics in college and was frequently irritated by the "intuition before rigor" approach, so I have a personal point of view on this.

First, intuition is an unreliable indicator. While solving a problem, or searching for something new, you are guided by intuition, but it isn't infallible. Sometimes you get the feeling that something will work, get very excited, and then you try to make it rigorous and it falls apart. There is a whole category of cranks who think they have created a perpetual motion machine or a theory of quantum gravity and don't know enough math to figure out that they haven't, so they live perpetually in that initial feeling of excitement. So, intuition is exciting, but you develop the instinct of not letting intuition get too far ahead of rigor without checking in. And vice-versa, if you go too far with calculations and no longer have a feeling for where it's going, you start to feel like you should stop. Lectures have to respect this yoking of the two together. If you just give me a bunch of calculations, my brain will revolt because the calculations aren't motivated. Likewise, if you give me a bunch of intuition without rigor, my brain will revolt because the intuition isn't grounded.

Second, intuition develops through practice. If you want me to have an intuitive ability to use a mathematical idea, don't hide it from me! Like, if you want to teach a kid to play baseball, there's a lot of talking involved, but maybe hand them a ball first and let them hold it in their hands while you talk, feel the size and texture of it. Even better, just let them try to throw it, and work up to playing catch. The motivating, big-picture idea of baseball includes things like the thrill of winning, camaraderie, suspense, and post-game snow cones, but their eyes are going to glaze over if you start like that. You get a little kid hooked by putting a ball in their hands and letting them throw it. Some mathematical ideas are fun to build from the ground up, and you can discover the snow cones and the cheer of the crowd later.

The third factor is irrational, but deeply felt for some people. Even though we know and experience every day that rigor and intuition exist equally and symbiotically, each rather pointless without the other, it is expected to give pride of place to intuition, as a way of catering to people who didn't enjoy math in school or are insecure about their abilities. "The really important thing is the idea, the intuition... calculation closes the mind and dulls imagination... the real geniuses have creative breakthroughs, and we are never short of less imaginative people to clean up the unimportant details afterwards." That's how, as someone studying STEM, you learn to talk to people outside of STEM, and it's chronically irritating and demoralizing that you have to denigrate an important aspect of what you do. So you tend to be hypersensitive to that chauvinism creeping into a "safe space" like a class in a technical subject. Of course each STEM subject has its own approach to mathematical rigor (mathematicians and physicists notoriously don't see eye-to-eye) but we all appreciate the value of it.

Also I think your third point is important. If you only get the math, you can use it, and the intuition will come. If you only have the intuition, you're in a much harder position to get going and develop the math.