PRetty sure in NY you can't burn, but the DEC does every few years on a miuntaintop where there is blueberry bushes. They run the fire right through the open mountaintop, and the blueberry bushes recover.
IF they recover, im sure tougher trees n shrubs would recover without trying to kill them first. I'd guess anything pinky sized trunk and smaller might not survive. Guessing here, I'd let the stuff you hack n squirt stand. I could see a stump cut low might recover vs having a charred burnt end ontop of the herbicide.