Glenn definitely did some videos where he shows that at higher gains the differences between pickups gets vanishingly small. I think that he didn't say you couldn't tell which pickup was which but rather you couldn't tell when one pickup was switched for another which sounds the same but is pretty different. But for all I know, its Glenn Fricker and he said exactly what you said he did.
There's a close parallel in mic preamps, where the sommeliers pretend they can totally easily tell even under the worst circumstances how much better a neve mic pre is from anything else, and then when actually tested, pick Mackies and ARTs as the best in some cases and in others cannot even tell IF mic pres had been switched beyond the 33% random chance would predict
To me, pickups have measurable resonant frequencies, and thus should be distinguishable at least measurably predictably, even if we can't or would have trouble hearing it, regardless of the gain as I can't think of any way that a multiband distortion model of gain nonlinearities could somehow even that out, but I think I remember the video you are talking about and it really did seem hard to tell.
And in my own experiments: I SWEAR that an EMG81 sounds like an icepick to the brain with its ceramic pickup, compared to an Alnico 87. I could hear it from a mile off...But when I tried, with two different guitars, thru a pretty high gain amp sound, I couldn't ABX between them
I'm very confused on pickups