Personally, I would stick with the straw. It's cheap. The sun hitting it will warm up area around it (because it's darker than the snow), and it will melt first in spots, making footing better. And in spring leave it to be ground into the mud by hooves. By adding plant matter to muddy areas you are (very) slowly raising the area, and the broken plant matter mixed into the soil absorbs the water better and will (eventually) reduce sloppy spring mud situations.

This is assuming you are talking about your muddy sacrifice area, not a pasture with good grass.