Carpet remnants will work as well or better.
I have debated snow tires for my corolla, but haven't pulled the trigger... I feel like the second I spend that much on the car, it's going to have a catastrophic failure and the tires will be a waste of money
I don't know where you live, but I drive 47 miles (each way) directly into Ohio's "snow belt." For those who aren't familiar with this term, my office is located near Lake Erie, so until the lake freezes over, cold Canadian air blows across the lake, picks up water, and dumps snow on Northeast Ohio even when there's no precipitation falling from the clouds.
Front wheel drive is really pretty great in all but the absolute worst conditions. I have only been truly stuck once. I also totaled a 2005 Nissan Altima in white-out conditions, after skidding into an existing 10 car pileup that I couldn't see until it was too late (I was doing about 40 mph. maybe less.)
My point is that snow tires help you STOP too, and that's probably more important for your safety than anything involving getting stuck.