Either way I am good with it...
I figured if it was breaking down then we'd pull it out and stick it in the compost to protect the flooring!
If it's not, no big, was just gonna go to the garden next yr
Sounds like you are good to go either way.
From what I see, you would be better off using that litter in the compost or using it as top mulch out in the garden. If you till those wood chips and shavings directly into the ground, they will rob the soil of nitrogen to break down the wood. That will result in less nitrogen going to the plants in your garden.
Wood chips make great top mulch in the garden for water retention and to block out weed growth. I use them all the time for that and it works great for me.