Good idea. Did that once, was an eye-opening process.
But we should first agree what we're chasing.
If we're looking for simple saturation or "linearly coherent" harmonic generation, there's not really much use for the transformer - this can be done at resistor-diode-capacitor level, simpler and cheaper. And yes, even including some of the first-order "tape saturations", that mainly saturates low frequencies (emphasis / deemphasis wrapped around a clipper) and filters distortion products in the highs. Easy and cheap enough, even in digital.
But the harmonic generation in the above "predictable" type of circuits is probably not what we really want, is it? It's usable alright, and even convincing at first - but, at least to my ears, it gets boring quickly. In this regard, the linear harmonic generation is no different from what is done in (recent, good) digital simulations.
What I like in analogue is the introduction of the slightly chaotic component. Not chaotic like noise, but chaotic in the sense of being non-predictable in the particular despite being generally predictable on macroscopic level of working. In other words, it (nearly) always does what you want and expect - it's just that you can't/shouldn't-be-able-to predict in each instance
exactly what will happen. Outcome will be distributed in ways or patterns that we can think of as attractors instead of determined. The chaotic component itself stems from miniature (too small to control) variations of initial conditions resulting in rather large variations in outcome, messing up perceived determinism (note that it's still 100% deterministic, just outside of our ability to control and predict).
From outside, feels like magic. Perhaps "organic" more precise
So, where to get that?
The simplest chaotic system consists of an inductor and a diode
"Chaos in a diode" (Andrew Missert and Peter Thompson):
https://www.pas.rochester.edu/~advlab/reports/thompson_missert_choas.pdf
"Chasing Chaos with an RL-Diode Circuit" (Junaid Alam and Sabieh Anwar, 2010):
https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwj14beZ6NvsAhVSzIUKHTOOCv0QFjACegQIBBAC&url=https%3A%2F%2Fadvlabs.aapt.org%2Fwiki%2FFile%253A2597&usg=AOvVaw00nJUX48SORHur3wjwzJlm
I haven't pursued this all the way down the rabbit-hole, but to me this direction of explanations makes much more sense for describing the subjectively experienced behavior of certain transformer-based circuits that we like a lot. Like
"no, it's not happening in the tube, nor in the transformer, so what is it"?
For the hardware, may I suggest the humble Monacor LTR-110 as a candidate? Good core size for our opamp-driven endeavors, many many different winding ratios possible, Core material of some crappy type with just the right amount of Barkhausen grittiness at low levels like reverb tails
/Jakob E.