if there's room to pinch it then the forktube becomes slightly smaller, making it easier to get out. cutting it puts like a millimeter or so of room in the fork to pinch, making it less snug against the bars.
wen my forks snapped they napped like at the slits and i just bent my slits a little with pliers and banged them on the ground n they fell out but if the bit of fork doesnt rattle or get in the way of ur new forks id leave it or push it all the way up