You do really need hay. The rabbits will often try to burrow or cover the rabbits with the nesting material, and if you only make it out of softer material, such as carefresh bedding, the nest can't hold its shape very well. I've tried it before; and the does seemed very disappointed, carrying around a mouthful of bedding and trying to build a nest out of it. What you can do if you want something softer than hay is to look for a different type of hay - instead of the more expensive timothy hay you can look for bermuda or orchard grass hay (check feed stores) which is often softer. You can also use a combination of the beddings - what I do with my rabbits is fill the nest box up halfway with bedding, then the top half with hay. That way the doe can build a nest out of hay but the rabbits still have the soft, warm, absorbent bedding under them (plus the mother's fur).
I have never, in over a decade of breeding, seen a kit seem to be bothered by hard hay was. Also - yes, hay in the nestbox does make an absolute mess, it gets everywhere. That's just the nature of breeding rabbits. It's most messy when the doe is nesting, after the litter is born and settled in she stops messing with the hay so much and most of it stays in the nest box.