Part II. Many of these amps/tubes have a "break-in". I kinda rolled my eyes but after playing my new tubes several days, I am almost almost broken in. They do sound different. I hear pink noise, or music. What does just leaving the amp on do?
Leaving it on has exactly the same effect. Don't listen to audiophoolery, it will only fill your head with nonsense.
The whole concept is, at best, highly dubious, and, at worst, proof of confirmation bias.
It is possible that some tubes may not have been fully activated by the manufacturer. The heat from the filament would, of course, fully activate any cathode material not yet activated, complete various chemical reactions, drive out any absorbed/adsorbed gas or water to final activation with whatever getter material remains, and generally finish what the activation process began. It isn't a burn-in or break-in, it's simply the tail end of a curve which already happened.
I don't believe that anything would be required. If it were, the manufacturers would have aged those tubes longer than the manufacturing process did prior to packaging and shipping, both to fully activate the tube and to weed out any quality control issues.
Assuming any issues existed, and likely none do, tens of minutes to a a few hours should be sufficient to resolve issues for any ordinary signal tubes. Transmitting tubes have a more complex process involving gradual voltage increases, but that does not here apply.
But, again, this is just oft-repeated dogma not having any proof. It's always "to my ears" which really means "to my adjusted brain".
Get a new mattress. Experience same effect. Did the mattress change? No, your brain and musculature adapted.
This is not the same as breaking in a mechanical device, like an internal combusion engine or a loudspeaker surround.