This. A thousand times this. It takes like three weeks of screwing around to get something remotely valuable, and oftentimes a summary takes just a paragraph that people look at and say, "Well, yeah, duh, that's obvious isn't it?"
Only then will you truly grok Feynman's joke about mathematicians: "Mathematicians can only prove trivial theorems, because any theorem, once proved, is trivial."
(When I first read that at age fifteen or so, it was humor. Now it is dark humor. Still funny, of course, but with richer overtones.)
Only then will you understand how James Clerk Maxwell's life work can simultaneously be one of the top ten intellectual feats in the history of mankind and something that most undergraduate physics majors learn in a semester or two.
(I'm speaking from experience.)
But it is true that we routinely use the equations to solve problems that were very difficult intellectual struggles for Maxwell. A big problem is that Maxwell was encumbered by his mathematics and notation - everything he did was later refactored to be more straightforward. The dev, grad, curl stuff we use today came later. We have Oliver Heaviside - a man whose name is now more famous from Cats lyrics - to thank for them.
Another, related, effect of most people not being exposed to engineering is a belief that engineers don't make gadgets do certain things simply because it hasn't ever occurred to them (i.e. ideas are everything, and implementation is probably trivial in comparison).
[1] Replace "Duke Nukem Forever" with your favourite overambitious project.
If you do any amount of engineering or building, you will learn a lot about how the process works. On the flip side, if you have never tried to engineer anything you'll be prone to magical thinking. Too many people's mental model of engineering seems to be drawn from Green Lantern comics: If a gadget cannot do a certain thing, it must be because its builders don't have enough will to succeed. They need to want it more!
The extreme example of this problem is in science. Just about nobody in society has actually done any experimental science. Being a fan of science, reading lots of books about science, visiting science museums, or conducting demonstrations of basic procedures doesn't count. It doesn't teach you what it's actually like to explore a problem where the "right answer" is not known in advance, and every data point costs money or time, and there are more knobs than you can afford to turn, and it's hard to resist cherry-picking the data, and your first N hours of work have to be thrown away after you find the boneheaded systematic error.