Broil is an ancient word of uncertain origin but certainly not American; most probably French. It was used with the same meaning by Chaucer in the prologue to The Canterbury Tales around 1386, long before America was colonised although he wasn't first. Shakespeare also used it in 1613.
Quote2.2 spec. To cook (meat) by placing it on the fire, or on a gridiron over it; to grill.
c 1386 Chaucer Prol. 383 He cowde roste, sethe, broille, and frie‥and wel bake a pye.
He could cook, roast, boil, broil and fry ... and well bake a pie.
I seriously doubt a truly exhaustive study of the regional uses of any of these words is feasible. Different people in some city apartment blocks could use all three to decribe the same process.