To be honest I think your explanation misses the point in two important ways.
First, the Fourier transform does not just measure the 'energy' or amplitude of a single wave; it must also take into account its phase.
Second, complex exponentials are in no way an essential ingredient for defining the Fourier transform. We just work with them for computational simplicity, essentially because exp(ix) exp(iy) = exp(i(x+y)).
I also have trouble assigning meaning to your last paragraph. After all, complex exponentials can cancel just as much as sine waves (proof: sine waves are a combination of complex exponentials).
I agree with these criticisms. Really, my goal when I talk about Fourier Transforms is to avoid talking about phase. It's important, clearly, but it's both less intuitive and less practically meaningful. For a lot of applications, the phase data is just discarded anyway. And if you're in a situation where it matters, you're probably reading more than 3 short paragraphs.
I'd love other thoughts on how to handle the ideas in the last paragraph. I'm not being technical, but am trying to allude to how the complex exponentials can capture more information more conveniently... and honestly waving my hands a lot. Really, I just want to explain why they're there instead of a more recognizable sine or cosine functions.
I've also tried to show the Fourier transform as the sum of a sine and a cosine transform, but that's too much, I think.
First, the Fourier transform does not just measure the 'energy' or amplitude of a single wave; it must also take into account its phase.
Second, complex exponentials are in no way an essential ingredient for defining the Fourier transform. We just work with them for computational simplicity, essentially because exp(ix) exp(iy) = exp(i(x+y)).
I also have trouble assigning meaning to your last paragraph. After all, complex exponentials can cancel just as much as sine waves (proof: sine waves are a combination of complex exponentials).