Mathematically derived waveform vs. captured waveform from an analog synthesizer
the individual parameters on an analog synthesizer could be controlled digitally.

OK so far.

instead of using mechanical potentiometers to adjust some quality of the waveform (eg the frequency of a LFO), use digital potentiometers.

It seems you already know what you want the frequency, amplitude, and phase of each component to be in digital form.  If you have all digitally, then it doesn't make much sense to convert those back to analog control signals, then produce the components from those.  That would introduce significant errors.

Even a modest DSP can generate sines with controllable frequency, amplitude, and phase much more accurately than a analog circuit.  Use enough bits, and you can easily exceed the signal to noise ratio of any analog circuit.  16 bits, for example, gives you 96 dB.  Analog circuits can do that, but what about 20 bits, or 24 bits?  That's just another byte for digital, but takes analog from "quite good" to "impossible".