Of the two types of bindweed, hedge and field (white flowers, thick rooots for hedge and pink flowers, thinner roots for field) you can almost (say 95%+) get rid of hedge bindweed in a year with 100% light exclusion. Field bindweed would be massively weakened and subsequently controllable with a trowel.
So yes I would cut to the ground and mulch, preferably black polythene as bindweed sneaks out the overlaps of cardboard.