Take e.g. a core audio belief of burn-in. A relatively simple test involving two headphones and burning one in for say 100 hours - a time period where a host of audiophiles on Head-Fi, 6Moons and other BS-afflicted sites will report major improvements in the sound - then comparing both in a blind test will fairly conclusively put to bed whether burn-in is a factor.
Same with audio equipment and 'burn-in'.
No-one 'prominent' has published the results in 10 years of Head-Fi existence though. Best we've had is a carefully-worded article by Tyll Hertsens on his site muddying the waters further.
Burn-in is actually a believable physical phenomena, though. It's simply wear and tear (or "conditioning") of various components inside a driver.
The only reason why nobody has been able to conclusively show you any consistent data is because:
1) It's not possible to measure microscopic changes within a driver with the instruments we have now, so nobody knows for sure "how much" has changed within the drivers in those 100 hours. But you can ask pretty much any engineer, and they'll tell you that there should at least be "some" change.
2) With #1 in mind, the only way to really "measure" the change is by measuring what comes out of the headphones themselves. This can be done with head models, but again... due to limitations of measuring instruments, the most you can say is that with your particular setup, you can't really pick up any substantial change.
3) People hear differently due to their ear structure and so on. I don't believe you can say definitely that with the same pair of headphones, I would hear the same way you do. I can probably pick up the same information, but it'll probably vary slightly to drastically. In that case, I'd think "burn-in" would be more on a per-person basis rather than anything you can scientifically prove for the masses.
So that's why I believe there hasn't been any definitive proof that "burn-in" is real or not.
I can agree with confidence, though, that under normal listening conditions, I haven't actually picked up any difference in the sound signature of a pair of headphones, and my headphones are all those reported to have substantial changes after burn-in.
So personally, I would agree with you that "burn-in" as a mean to substantially change the sound of a pair of headphones does not exist, but I am also open to any report that tells otherwise, because I know from physical intuition that there must be some change.
As a lowly BS Computer Engineer, I agree, it isn't that complicated. I've seen TONS of things come from audiophiles and their ilk that have caused a little part of me to die.
"That HDMI cable has to be gold plated for a richer sound!"
"Vacuum tubes produce a much richer sound than their digital counterparts."
"Records are much better than any digital storage method"
999 times out of 1000 they are completely full of crap.
Depending on the situation, though. I don't think it's a full 999 times. There are some truths to it:
1) The HDMI cable... totally agree with you there. The fact is that it's transmitting digital data, not analog, so any signal loss is unsubstantial unless the receiver is totally crap.
2) Vacuum tubes have richer sound than digital counterparts (OpAmps). Generally... pretty true, actually. Fundamentally, tubes and opamps do things differently. Or in fact, different opamps do things differently. It also depends on the circuits as well. What I can definitely agree with, is that in general, it's not true that vacuum tubes would produce better sound than opamps. But that's really something up for debate because it's purely subjective. And there's also the fact that in general, vacuum tube amps are better received (and genuinely sound better) than digital amps due to design decisions and other factors... so it's really hard to tell.
3) This I can say... it depends on what the digital storage method is. Original 96KHz 24-bit PCM files would probably be on par or actually better than records. But if you compare MP3 320kbps to records, then... I think it highly depends on what song it is. You may disagree, but I think music stored in MP3 128kbps would be obviously and inherently inferior to records.