Second thing first - The sharpness of the leading and trailing edges of a squarewave do not cause "ringing at any and all frequencies." You are confusing a periodic wave with an impulse, or infinite impulse in the FFT world. That has energy at all frequencies, but only because it is not periodic. Leading and trailing edge sharpness *do* affect the shape of the harmonic field. The harmonic amplitudes defined by Fourier are theoretical limits; real world waveforms have lower amplitudes because the edges are not infinitely fast.
BTW, a triangle wave has no fast vertical edges, yet has same number of harmonics as a squarewave.
First thing second - Your view of how individual waves combine assumes that the rms values add linearly. They don't.