There's a lot of great explanations here, but none that quite put it the way I'd think of it.

If we had primary color wavelengths that could stimulate each cone independently, then it would work just like you say, and we'd only need 3 of them. But because the cone spectra overlap, we don't have "orthogonal basis vectors" to work with. Our primary colors each excite a mix of cone responses.

But no problem right? As long as each primary color has a different response, we at least have linearly independent vectors, and any student of linear algebra knows you can mix those together to act as an orthogonal basis and get any desired excitation of the cones. Right?

And that would be true, except that linear algebra assumes you can freely add or subtract vector amplitudes, but with LEDs we can only generate light, we can't send a beam of "negative green". So we're constrained to the subset of colors where the basis vectors all have positive amplitudes. And that's the smaller color space that results.