If it is steep and icy enough for crampons, then I would want to carry an ice axe and know how to use it to self arrest. If you either don't have one or have one and don't know how to use it and you fall on something steep, you could find yourself in big trouble in about 3 seconds. I took a mountaineering class many moons ago, so my practical skills are pretty much nil at this point, but I do remember how quickly you pick up speed when you fall., especially with a pack on.

Another thing, if you do fall and try to stop yourself with your crampons as you slide, you could find yourself cartwheeling down the hill and risk breaking an ankle.
Don't get me started, you know how I get.