You will need an MTB mech for a 32 sprocket.
It isn't anything to do with the cage length. It's because the angle the parallelogram is set at isn't so steep for road mechs, and the top jockey doesn't move away from the axle so much as the mech changes towards low gears. If the mech does manage to shift the chain on to the big sprocket, the chain will be pinched between the sprocket and jockey, and it will run with a fair bit of noise and vibration.
Cage length is to do with how much spare chain can be dealt with as you shift onto smaller chainrings or sprockets. If the cage isn't long enough, the chain will be slack on small/small, and you will get the chain running across itself at the top jockey where the end of the cage has gone past the horizontal.