Don't know about your food processor but the ones I have seen are not just a plastic cylinder but also a funnel and circular blade attachment plus a couple of other bits that all tend to get pieces of food on them.
It's not impossible to clean and worth it when making larger amounts but definitely more of a hassle than a knife, even with a dish washer.
Yes, I agree it's more hassle than a knife. I don't agree that it's more hassle than a box grater.
The box grater (in the biscuit dough case, just for the butter) fits into the top rack of my dishwasher along with the cups.
It is.. at least my box grater which can be folded out flat making it very easy to clean the inside with a sponge and a tiny amount of elbow grease (much like the outside).