Perhaps American English was influenced by Dutch, in which "Rooster" means "gridiron, grill, griddle" (akin to roaster). Perhaps a rooster was a chicken fit to be roasted—as opposed to a cock, which was kept for other purposes.

From A New and Complete Dictionary of Terms of Art by Egbert Buys (Amsterdam: Kornelis de Veer, 1768), page 199:

To BARBECU'E , (Ind.) to dress a Hog whole, by splitting it to the Back-bone, and broiling it upon a Gridiron. Een Varken langs het Ruggebeen ophakken, en heel op de Rooster braaden.