However, the problem here is that we are not born with an innate set of fixed intuition and common sense. They evolve and expand as our knowledge increases. And that's where we run into trouble with QM, because almost every aspect of QM is counter-intuitive. There is an obvious discontinuity between "what we already know" and "what QM is saying".
It is the mathematical formalism that saves us from dangling in mid-air by providing a link to forms we are familiar with. Once we understand this, we then reformulate our "intuition" based on new set of ideas and allows our intuition to grow and encompass this new set. But it comes out of an understanding of the mathematical formalism first and not simply an expansion out of a current intuition.
It is due to this that I disagree that one can go far in learning physics simply based on intuition alone. There's a very BIG roadblock in the form of Quantum Mechanics.