I had similar confusions; despite years of engineering classes, I didn't build an intuition for Calculus until a decade later. It felt like such a waste. (Yes, I could memorize rules; but could I visualize the product and quotient rule?)
I decided to Elon Musk It™ and reason from first principles (http://betterexplained.com/articles/honest-learning/).
What would a learning plan look like, assuming limited motivation, a need for genuine understanding, and goal of starting with appreciation and moving to proficiency?
Just like learning music, I'd listen to the song, then clap along to it, then play a few chords along with the band, then learn a bit of music notation, and then practice playing it. I wouldn't start with scales (er... limits) for weeks on end. Not if I was honest about my motivations.
Ultimately, we have to acknowledge the approach that helps us. For me, investing a few hours of exploration before the formalities is well worth it.