While i see where you are coming from; i feel that you are putting the cart before the horse. While Intuition by itself is not enough, it should absolutely be the first thing you should focus on before doing the hard work through rigor and formalisms. The former can be "grasped" while the latter needs "practice and applications". This is how Science itself developed (a good example is Faraday vs. Maxwell's approaches). Intuition/Rigor are analogous (in a certain sense) to Theory/Practice. You need both, each amplifying the other's effects at various stages.

Here is a neat communication from Faraday to Maxwell on receiving one of Maxwell's paper;

“Maxwell sent this paper to Faraday, who replied: "I was at first almost frightened when I saw so much mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well." Faraday to Maxwell, March 25, 1857. Campbell, Life, p. 200.

In a later letter, Faraday elaborated:

I hang on to your words because they are to me weighty.... There is one thing I would be glad to ask you. When a mathematician engaged in investigating physical actions and results has arrived at his conclusions, may they not be expressed in common language as fully, clearly, and definitely as in mathematical formulae? If so, would it not be a great boon to such as I to express them so? translating them out of their hieroglyphics ... I have always found that you could convey to me a perfectly clear idea of your conclusions ... neither above nor below the truth, and so clear in character that I can think and work from them. [Faraday to Maxwell, November 13, 1857. Life, p. 206]”