So... theory and experience asides, based on experimental evidence, why does swivelling the barrel produce a near immediate increase in activity?
"increase in activity" is granted based on those "secondary indications" you mention as "unreliable".
However, if all other things remain static, the airlock is not bubbling, the barrel is not under pressure, and you swivel the barrel, the airlock starts to bubble and the barrel looks under pressure.
Also, the first 2 brews I did, I left them still for the entire time, had the intense first foaming with puffed up lid, but promptly slowed down to virtually nothing happening. 7 days later and the FG was higher than it should be by a margin.
The first one I swivelled every few days, if it looked slow, and 7 days later and an FG lower than expected and nice smelling, clean beer. Although, I have yet to condition it, it tasted better than the first 2 out of the barrel.
I know the party line. Leave it longer instead. Still doesn't disprove that gently disturbing the ferment seems to speed it up.
I'm thinking it's not about disturbing the trub or waking up the yeast, but more to do with the yeast colonies not being spread evenly throughout the wort, if left alone you have to wait on them growing through the beer. Create an occasional current in the wort causes the active colonies that may be running out of sugar where they are to spread through out the wort and work better on more sugar rich wort again This is all conjecture though.
And, finally, sure, the party line advice of "Just leave it longer, if it's slow and takes 4 weeks, it's just slow and takes 4 weeks", will still work and will probably lower the risk of off flavors.