Can harmonics be created or simulated a la carte
Without re-hashing all of the maths from RBJs response, I made a crummy little plug in that did more/less what you are proposing. It had a stupid simple GUI with one slide for nominal amplitude and several more to control harmonic levels. It was OK, but not as useful as one would hope for audio work. The idea was that given a sinusoid with a nominal input amplitude, it would calculate the polynomial required for the given harmonics.  The bummer is that audio doesn’t work that way, at all.  The sound doesn’t have any more harmonic ‘richness’ if you can call it that, it just sounds distorted in different ways.
To wit, my peripheral understanding of products advertising harmonic distortion use crossover filters to divide the signal into different bands, on which distortion is applied. By narrowing the bandwidth, you get something more closely approximating a sinusoid. Again though, my impression was that they were fine, but not exactly what I had in mind.
My experience was that as simple as ‘adding harmonics’ seems, it’s a pretty tough thing to implement. My apologies for not providing a more rigorous analysis, but figured some first hand experience might be relevant.