Generally speaking analog circuits can be pretty well simulated with a circuit simulator, like the freeware ltspice. I mean, I don't think there is black magic behavior in transistors, tubes or transformers that can't be expressed in a circuit-level simulation model.
I don't have a "golden"object or patch available, but here I'll try to break down (guitar) distortion into parts that may or may not be a relevant part of its desirable character.
A common approach to modeling distortion is simulating the amplitude distortion, like a waveshaper. A "naive" sample-by-sample digital implementation a waveshaper often suffers from aliasing artefacts. Combating aliases is an important requirement when modeling high gain distortion. Current objects that have some sort of strategy to reduce aliasing are "dist/inf", "dist/rectifier", "dist/rectifier full" and "dist/hardclip", and the newer "jt/dist/class a".
This video shows quite some fuzzbox waveforms on an oscilloscope, I mainly observe waveshaping behavior, and it seems pretty close to hard clipping, not much evidence of a soft transition between clipped and unclipped or frequency-dependent behavior.
But remarkable is that the clip levels seem to shift with the amplitude, at this point, you can see that over 180 degrees of a sinewave cycle is clipped away, meaning that the positive side clip threshold moved to a negative level. So the operating point of the waveshaping is dynamic, if the positive side clip level had been static at a negative level, it would gate small signals.
Still waveshaping is not always a sufficiently complete model.
Looking at the schematic of a tubescreamer, there are not just 2 clipping diodes in the feedback path of the opamp, there is also a RC network to the virtual ground, adding some filtering into the distortion.
Also some gentle high-pass filtering before distorting is relevant, I have noticed guitar pickups can produce substantial subsonic output by just pushing the strings near the pickup. Without filtering, this would change the operating point of the distortion.
In a guitar tube power-amp, producing loud output may "starve" the powersupply, causing the supply voltage to drop when you hit the strings, called "sag". This changes the operating point of the whole circuit, adding a sort of compression. I have also heard anecdotes of guitar players preferring the sound of a nearly empty battery in a tubescreamer distortion pedal. Something similar can happen there. This video demonstrates a potentiometer between battery and a tubescreamer to control the sag. Remarkable is the production of subharmonics, while a waveshaper can only add overtones. I think (subtle) addition of subharmonics are desirable. And I imagine, when a tube amplifier power supply is sagging, it also increases the amount of mains hum. But I haven't heard of anyone using a mains frequency converter to tune the hum to be in harmony, or bands tuning their instruments consciously to mains (requires convincing the other band members...). For 50Hz mains (Europe), hum is between G and G#, in the US, 60Hz, between A# and B. I also haven't seen any guitar tuner that offers mains frequencies as an alternate tuning reference
Transformers do not just simply saturate, their response also has a sort of memory called hysteresis.
Typical guitar tube power-amps use corrective feedback, from after the transformer to the input of the poweramp, reducing gain but increasing its linearity. But as this happens across all the effects listed above, assuming its effect is only linearizing is probably an oversimplification.
Not every effect necessarily has a substantial or desirable contribution to the final sound, not everything needs to be modeled to obtain a satisfactory model. A good synthetic test-waveform to play and record through "test" pedals and amps that reveal specific properties should be designed. For instance, a pulse is invariant to waveshaping, and will reveal the frequency response of the system after the distortion. And low-amplitude pulse mixed to a low-frequency sine wave can reveal the small signal impulse response at different operating points by varying the phase of the pulse and the sine wave. Then, the recorded wave could be analyzed with Octave or Scilab. Is anyone aware of such an (open source) effort?
But even without further scientific analysis I hope this breakdown could be inspiring in patching up distortion techniques. And if it sounds good, it is good.