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.