I think there's merit to both sides of the discussion... I've certainly added and subtracted elements from my system that improved and or worsened the sound of that system. In the same breath, I've also changed components that I perceived as everything from strangely different, to down-right unpleasant, that subsequently became favorites, even cornerstones of my favorite set-up.
Do I think there's a mechanical concept for burn-in? Depending on the component, yes I do. I would be willing to swear in a court of law that tubes improve from new to 100 hours or so. I personally don't like the sound of a brand new stylus, much preferring one that has 50 to 100 + hours on it (and that's a stylus bought to replace an existing one of the same model).
Capacitors? I dunno, I vaguely understand the concept of "forming", but I couldn't say with any degree of certainty if the "improvement" in speakers I've recently recapped is anything more than my ears gettting used to the new crisper, sound. I CAN say that recapping an older speaker will definantly alter the sound I've grown used to, but once I've listened to it for a dozen hours or so, my memory of the "original" sound grows so dim that I would be hard pressed to recall what they formerly sounded like.
This is one of those questions that IMO cannot be quantitatively answered. In the end the individual listener must be the final judge. No mechanical measure can determine (yet anyway) that a piece of gear will be pleasing to a particular person. I feel sure that each of you can think of at least one piece of gear that has a huge following, that you don't like, and wouldn't own. It doesn't mean the piece of gear is crappy, just that you don't like it, and that (in the end) is all that matters.