I read a lot of the typical Bike Forum solutions to a problem: spend lots of money on buying specialized equipment and gear because that has to be right. Well, maybe it is, but I don't have lots of money to spend on anything. However, I have lived in Wisconsin for 60 years and we do have winters here--very cold winters with snow and ice. I have no car so in the winter it's the bike and cargo trailer for everything and I've learned how to stay warm, including my feet.
Now I have worked outside for years before I retired and I even have a pair of boots that are rated to -100 F in case I ever do an Antarctic expedition. But riding my bike in the winter I wear a wide athletic shoe maybe a size larger than normal and 2 pairs of Merino wool socks, the greatest socks ever made for keeping your feet warm and I wear them all winter whether riding or walking.

Last year after a snow/ice storm I couldn't ride my bike so I walked and pull my cargo trailer to the fire station to fill a couple of 5 gallon buckets with free salted sand. The sidewalks had not been cleared or salted so go back home I chose a different route that was easier when I got to the driveway that lead into the apartment complex by me and there was a puddle. The trailer was a bugger to pull by hand with the full buckets so I decided to walk through the puddle since I was only a block from home. It was over my ankles--cold! cold! cold!. But even though my feet, socks, and shoes were soaked, after about 10 yards my feet began to warm up because wool retains its insulating properties even when wet.
But whatever you choose to wear on your feet on the outside, use Merino wool socks on your feet.
Ironically they are also good in warm weather because they wick the moisture away from your feet and as a bonus, they don't smell. Plus, they are not expensive.