What exactly destroys the RR when there is no battery?

I think you understand the operation of the regulator quite well.
I don't think the regulator is going to have any problem with the unstable voltage.
Depending on the capacitance load of the avionics, It will either oscillate from near zero to the full output which can be 50+v or it will just shut down as you suspect.
The problem will be if the avionics can't tolerate 50V Alternating DC at tens to hundreds of kilohertz.
I would be surprised if the Regulator died, but losing one or more of the avionics packages would not be out of the question.
I do not suggest trying this at home but I suspect there is a bit of Old Wives Tale involved here.
You are not supposed to ask WHY; You are just supposed to do as instructed Grasshopper.

The frequency wouldn't be in kilohertz, probably no more than 800 Hz! Because the SCRs once triggered will stay on until the sinusoidal voltage drops back to zero, so they can't be pulsed any faster than the AC frequency. It doesn't have a full-bridge rectifier and then smoothing, and then regulation by pulsing. The rectification and regulation are done together (It's dual half-wave rectification done by two MR2510 diodes and the two TP154E SCRs, and the regulation is performed only by the SCRs).

So at maximum output, the SCRs would be triggered immediately (minimum output would be not triggering at all even at the peak of the sine). As I look at the schematic, it looks like it's most similar to a triac-based light bulb dimmer. It isn't turning on and off, but it's triggering the SCR earlier in the sine wave, letting more pass through. One of the image attachments shows this, except with a potentiometer instead of the complex circuits for sensing voltage on C. Is that correct? Or does it not adjust its trigger point at all and only lets through a wave or blocks it entirely?

A low capacitance load would damage avionics like you say, but I kept hearing that the Ducati could be damaged and that's why the big capacitor is needed for LFP batteries (to protect the regulator from self-destruction if the BMS isolates the battery). One of the selling points of the Silent Hektik is that it is not damaged under no load (I think this is because it is a hybrid series and shunt regulator and if the series regulator isn't able to reduce the voltage by pulsing, then the shunt kicks in).

In the first attachment for the Ducati schematic (found on https://www.circuitsonline.net/forum/view/135679), if you get that pulsing 50 V on the C pin (labeled "+12 volts from headlight circuit") would subject the 220 uF capacitor and the Q1 transistor's base to high voltage, but Q1 is a generic high-voltage PNP. I don't think any of those components would be damaged.

When I get my hands on a spare Ducati that I don't mind never putting back into a plane, I'll hook it up to a variac and oscilloscope and experiment with its failure modes under various loads.

A forum member on an Italian forum claimed that from experience, "senza carico il regolatore muore in pochi secondi" (translated: without a load, the regulator dies in seconds). I think I might try this while connected to monitoring equipment and then open it and see which component failed.

As the voltage on the C pin rises the wattage on the resistors and zeners in the circuit also increases. Above 20 volts the applied wattage exceeds the component rating.