Some people hear a jar of pebbles changing the way speakers sound. The most fallible sense is probably hearing...
The best way to demonstrate burn-in taking place, would be some kind of objective measurement, not using someone's ears...
As far as caps are concerned, my understanding is different caps construction involve some different mechanisms of how energy are stored and released, especially along the time axis and frequency band. Many caps involved an electro-chemical process which is far from 0 or 1 by definition. The micro equilibrium dynamics and revisibility of a chemical process is a black hole to me but I am sure there are more knowledge involved in this area.
I guess it will be difficult to measure, real time, what is chemically happening inside a capacitor, especially under transient of high voltage / current swings.
Another thing, before we close this off, is that objective measurement of "audio experience" as perceived by individuals have not made too much progress in the past 20 years to my understanding. For example, how and why certain chain of audio nodes and signals in different arrays and time space will make a certain person feel relaxed, happy, sad, excited, or irritated is still at hypothesis stage. The subject is made more complicated and non-conclusive by other determinants like the intelligence, background, cultural environment of a particular test subject. Objectivity can be a real challenge in this connection.
Yes, we can measure physical parameters like wave forms, voltages, currents, impedance, frequency response, and various transients at nano-level. We can even trace the path of a certain high speed particle in space, but it will be difficult to "relate" this parameters to audio terms.
Not least, no matter what parameters we are measuring on the bench, it is very likely our magical biologicial system does not intercept and interpret things in the same way. Our own decoding system is much more complicated, sophisticated, but less predictable than a oscilloscope
I believe science is constantly advancing and there are things that we cannot fully explain / decode yet. But that does not mean certain events are not taking place.