Technically the smoke ring is formed by nitrogen dioxide present in the wood smoke in reaction with myoglobin in the meat. The presence of smoke has everything to do with it, which was at the heart of the OP's intial question.
I took the heart of the OP's question to be "But what about the guys using the
AMNPS?" (his question in his second post) i.e., lots of smoke, but no smoke ring.
Technically smoke is a collection of airborne solid and liquid particulates and gases. NO2 is one of those gasses that is a byproduct of combustion, in varying levels depending on the type and conditions of combustion. As noted elsewhere, a gas grill heats by combustion, produces no "smoke" or smoke flavor (by a BBQ standard), yet can give you a "smoke ring" because of sufficient levels of NO2 as a byproduct of propane combustion. A internal combustion engine can do it too, but I'm not cooking my ribs over the exhaust of my F150, lol.
An
AMNPS gives you plenty of yummy visible smoke yet does not release enough NO2 for a smoke ring. You could expose cooking meat to pure NO2, get a pretty smoke ring, but have no
discernible traditional "smoke" flavor.
My point was simply, just because you see and smell "smoke", does not mean you will get a "smoke ring".
And just because you don't, doesn't mean you won't.
Now, that being said. The flavor profile of a butt cooked in a MES with an
AMNPS is not the same as a butt cooked in a stick-burner. Precisely because the stick-burner brings a lot of byproducts into play in concentrations that an
AMNPS doesn't. Not saying which is the "better" approach, but they are different, just like charcoal is different.
Don't get me wrong, the AMNPS is a great product, I love mine, and use it all the time. Canadian bacon, smoked salmon, smoked sausage, smoked cheese, etc., can't be beat. And and
AMNTS does a great job of adding smoke flavor in a charcoal smoker (but in that charcoal smoker, it's the charcoal, not the
AMNTS smoke, that will give you your "smoke ring").
They're all tools, and all have their place.