(a reversed Si diode across the LED to limit reverse voltage)
Can not have that, because the goal was to make a diode tester that uses AC, such that the DUT LED polarity won't matter for the tester. This requirement is for easy testing, so I won't have to reverse the probes if a LED doesn't shine at the first try.
Main usage for such a tester would be for individually probing the LEDs found inside the domestic LED light bulbs, they are already soldered in strings, similar with these:
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Even more, the Vf can be 3V..12V, because inside each SMD, depending on the model there can be 1 up to 3 series UV LED junctions (all inside the same SMD, and covered with a white phosphorus that converts UV to visible white).
Usually it's not easy to make contact with the already soldered SMDs, that's why the idea of a tweezers tester that will shine the LED no matter the polarity, and such the question about what damages a reversed LED. Even more, I would like the tester to be a standalone small handheld thing, battery powered, so I won't have to plug anything into an AWG or PSU.
And on top of that, sometimes there are paralleled LEDs too, for example I have a LEDs plate made of 5 series sections of 3 SMDs in parallel, each SMD with 3 junctions (so I have to have enough current to light up a set of 3s3p UV junctions).

Meanwhile, I did a few tests already with a Joule Thief (a single transistor oscillator, that produces HV spikes), but I don't like that it outputs an asymmetric waveform, which might make the same LED to shine at different intensity depending on polarity:

So I've come up with my own schematic:

This oscillator produces a symmetric AC waveform, so the DUT LED will light the same no matter how the DUT LED is oriented.

L1 and L2 are standalone inductances, not a transformer like in the Joule Thief. The exact value for L1 and L2 doesn't matter much. I've tried with 100uH instead of 3.6mH and it still works. The ones I've used in the end were 2pcs of 3.6mH, recovered from two broken LED bulbs.
Tested the circuit on a breadboard (so far without all those polarity indicators, only the DUT and the oscillator) and it seems to be working fine. Tried to probe 2 different types of red LEDs from the 70s-90s (thinking the old red LEDs are the most sensitive to reverse damage) and they seem OK after probing them shortly, for 1-2s.

For probing those 3s3p LEDs found on the LEDs plate in the picture, I had to lower the resistor in series with the battery, even though I didn't connected any of those additional strings of LEDs seen in the schematic, at the left side of the DUT. Those 18 additional LEDs are supposed to indicate the DUT polarity, and the DUT type (1, 2, or 3 LED junction inside the DUT), depending on which ones are turned off by connecting a DUT.
I'm not sure about the indicators. Even if I use LEDs with 2 or 3 junctions/SMD, there will still be a lot of extra LEDs to shine.