For frozen footprints and ice the fatbike wouldn't help, you need studded tires. The fat tires would help in deeper snow for sure. But when you are on pavement, it may be plowed anyway.

Find out what your city snow removal policy is for bike paths. But even with snow removed, there sill can be ice. I rather be stuck in deep snow than to fall on ice and break my wrist. Try studded tires on your current bike.

If there is a lot of salt and wet, use the oldest bike you can get.

Single speed sounds like a horrible idea, you need lot of torque to stamp through deep snow, and want to go faster when the path is clear.