Yep, there is a fundamental tension between intuition and rigor. Unfortunately, I feel that rigor (and not even rigor -- rote details) are overemphasized, without giving any context, and more importantly without giving any inspiration to the student. It's like focusing on the spelling of a poem, not the content.
I do want to jump into the rigorous definitions in later posts (such as when limits exist, when they don't, why it's important) but I find intuition -> rigor is an easier path to navigate than rigor -> intuition [which often becomes rigor -> memorization]. If someone had just said "Hey, limits and infinitesimals help us solve the problem of 'How do we make an approximation which is still accurate enough?'" calculus would have clicked a lot faster for me :).