tlw
I've a Korg tuner with claimed accuracy of 0.1 cents and Peterson's strobe tuner software which is possibly even more accurate.

In reality they both wander slightly as the string tension changes following even a gentle pluck.

Melodyne is great for showing the extent to which this happens because it displays the pitch center as a line within the "blob." Note how the pitch center starts off about 20-30 cents sharp on the pluck, then takes a while to settle down into a constant pitch. This string actually isn't that bad, it's a D string on a guitar. Bass typically shows a wider excursion over a longer period of time. I drew a line in green to show the actual pitch center.
 

Anderton
Yes, but that's not really the point. You can tell there's a difference, but you probably will not be able to discriminate the pitch difference.

While I broadly agree with you, consider this. When tuning accordions with two voices in the treble (two banks of reeds tuned to the same octave) it's conventional to tune one bank to datum and the other very slightly sharp (typically anywhere from 5 to 10 or more cents depending on amount of tremolo wanted and the pitch of the note).

The reason the "tremolo bank" is tuned sharp is because if it's tuned flat the instrument might sound OK on its own but tends to sound slightly but noticably flat, especially when combined with other instruments (or even against its own chords and bass). The same applies to many tremolo harmonicas.

Whether it's something peculiar to free reed instruments I have no idea, but it's noticeable enough that the better accordion makers stick rigidly to the "tremolo bank" of a 2 voice instrument being sharp. I suspect it's a psycho-acoustic thing and slightly sharp strikes us as less out of tune (or brighter perhaps) than slightly flat. Kind of the same psycho-acoustic phenomena that make pianos sound more in tune when stretch tuned - which means a big chunk of their range theoretically isn't in tune at all.

There has to be a reason for this, and I'm not convinced that tests involving abstracted listening to pitches outside musical context really tells us that much about real world instruments.

Whatever, one things for certain. If the band/orchestra decide to tune to A=397 or 443 or even 10 Hz either way of 440 no-one is going to notice.