It is possible for a plastic barrel band to affect accuracy especially if the barrel is getting hot. Though there is a good chance the barrel band has nothing to do with it.
There is a couple of things happening to the barrel to consider. For one when the barrel flexes a bit when the pullet goes down the barrel, what you want is the barrel to flex exactly the same way every shot.
The other issue is the structural tension the barrel, a lot of barrels are made from rolled sheet metal and they have a now invisible seem in them. The seem has different characteristics then the rest of the material in the barrel. As the barrel heats up the barrel begins to bend toward or away from the seem. Even barrels that are not made from rolled sheet metal can do this that is why processes like Hammer Forging, Drawn over Mandrel and Cryogenic stress relieving are down on higher end barrels. Having a thicker walled barrel is a nice way to have a barrel that flexes or distorts less, of course it take more material and is heavier. I am not sure how a factory 10/22 barrel is made.