Bunnies like to poo while they eat- so that is often why hay is in or near the litterbox- to encourage litterbox use.
I don’t like to fill my bunnies litterbox with hay- I put it in a bin right above it. If they pull it out and put it in the litterbox I figure that is fine by me- but I feel bad basically serving their hay out of their toilet (litterbox). Schroeder always immediately digs his out into the box- and I can live with that- but I like to give them the option of “unsoiled” hay- although they do not eat any that is soiled with urine I am sure.
I use fleece blankets for them since fleece is easy to clean and cheap and most bunnies cannot get into too much chewing trouble with it. Shavings are messier and more expensive in the long run. And you want their cage bedding to be distinctly different than anything that can be confused for litter/potty area. My bunnies can tell the difference between their litterbox with woodstove pellets and the pen lined with a fleece blanket. There have been occasional accidents early in training and there is an occasional what I suspect “overshooting” the edge of the box- but since the blankets are theirs I don’t care if they end up looking a little “moth eaten” (from nibbling) and I change their blankets 1-2 times a week anyway to keep up on any hay mess/dust anyway.