However, the diodes (and the paralleled cap.) are inserted in the opamp feedback loop, instead being from the output to ground, as in the schematic on the initial post. I'm not sure if the opamp circuit will work well on a preamp tube circuit. Additionally, this parallel capacitor will have an increased effect on the next tube stage due to the Miller effect.
In the pedal world there is "soft" and "hard" clipping. The former typically refers to antiparallel diodes in the feedback loop of an opamp. The latter typically refers to antiparallel diodes to ground (or some other reference voltage functioning as AC ground).
You found RG's technical dissection of the TS circuit (probably the most famous soft clipper).
For an example of a hard clipper take the DOD Overdrive 250/Dist+ (or even a Klon for a more advanced application of this principle). I've included a screenshot of a schematic with the relevant portion highlighted.
The cap can be employed in either config, and since it's on a switch in your implementation the Miller capacitance effect is only happening when the diodes are engaged--which you might want in this case.
The Jose mod uses the diode function of transistors (whose internal capacitance effects might nullify the need for an additional parallel cap) or even Zeners for higher clipping thresholds. The principle is the same, clamping signal to ground for hard clipping. You can have fun with different thresholds for the positive and negative swing as well for asymmetrical clipping.
I'm not personally a fan of seeing this in an amp as I think it can be done better in the context of a pedal (with additional filtering, EQ, etc) but there is a whole world of possibilities with clipping thresholds, asymmetry, parallel caps, and series pots available for experimentation in an amp.