This is simply not true. Every display has a color gamut that is
limited by the maximum saturation of its primaries. You can
produce any color within that gamut but not any outside.

Correct.


Even if every flesh tone is in that gamut, that does not mean that
you will get the right flesh tones for a given combination of RGB.
In order to do so, you must have the same spectrum in the
primaries that you have in the camera filters...

This bothers me. It might be true in a practical sense, but it's always
struck me as being theoretically wrong (mostly because of the extreme
overlap of the eye's blue and green receptors). I won't start an argument,
though, because, even if my intuition is correct, I don't have the "science"
to back it up.