Feynman explained differential calculus in 6.2 pages, half of which was more focussed on physics, and he had plenty of time for a lengthy real world example. It's in his physics lectures, 8th one in the first book.
His explanation was about 5000x better than anything I heard in school. I conclude school is worthless. One can learn more in an hour of Feynman's lectures than a semester or two of school. Note also: schools could, but do not, use Feynman's lectures. Improvement is easily available and rejected.
(Perhaps you can find a school somewhere that does use his physics or computer science lectures. I've never heard of one, and have heard that they are generally considered unsuitable for a school course. Complaints are made about both the difficulty and the style. But both are simply superior to standard textbooks...)
I am not exaggerating this factor of 5000, though it's hard to make it a number. I was never given any *explanation* of calculus in school, despite several calculus classes (as well as some physics classes). So a more accurate number would be infinity times better. Note that I refer to both high school and university classes, and that university was no better than high school in this regard.
In school, they give you formulas for doing differentiation, and problems which you can solve using about 5 formulas you memorize. They never explained where the formulas come from, why they were invented, what they mean outside a few set examples, or how to reinvent the formulas from scratch. They made a very poor attempt to state what dx and dy mean (it's the derivative of x with respect to y). Feynman addressed that same issue much, much better.
Without explanations, memorizing is the only choice. And if you forget, you can't work it out again with pen and paper, you have to look it up. That is awful.
If you ask a teacher why they never explain it, they probably won't know what you mean or have thought about it. But once you explain yourself more, they'd probably say either that it's too hard, long and complicated to do that, or that the kids wouldn't be interested and don't need to know it.
But Feynman did it in 6 non-dense pages, so it's not too long, hard, or complicated. (Admittedly 3 or so of those pages may be dense to some people. For those people, normal math textbooks would be dense too. And anyway, 6 dense pages would be far shorter and easier than the curriculum schools use.)
As to interest, it's certainly more interesting than memorizing formulas for no reason, which the large majority of students already consider boring.