You seem to have forgot that going from FWD to AWD costs more than a set of winter tires. Plus with winter tires the summer tires last longer. So net, yes you will save money with FWD and winter tires.
I have also been on icy roads where no one could get going and I motored away in my Focus with Michelin X-Ice XI-3's. Prior to the winter tires that car with normal all seasons was the worst winter car I have ever driven. It was downright scary. The Flex showed almost as much improvement.