At least pizza boxes can go in the yard waste bin to be taken to the industrial composter. My yard waste bin came with a sticker that actively encourages putting food soiled paper in it.

You say that like that's a thing available every where. it's not. I compost my lawn clippings and food scraps, but I don't go so far as to put in cardboard type stuff in there. As for the greasy parts, it's usually just the bottom, so I've ripped off the top for adding to recycle and trashed the greasy part. But even that's more than most people would be willing to do

if only I had one in my area... where they exist use them, but they don't exit in a lot of places.