I'm in a similar climate as Indy. Smaller city, though (Springfield is 115k).

I have two short segments of bike path that I ride that are adjacent to streets on my route anyway, so when weather is good, I ride 'em. When the snow hits, it's like you described on those paths - the first ride is okay when the snow is wet, then once the tire ruts and footprints have frozen the next day, forget it. So snow events generally shut down those paths as options for me for awhile. That's okay, because they are short segments for me, and I can ride with traffic anyways.

Snow and ice can affect the streets I ride too, though. The city prioritizes the higher-traffic arterial streets for first snow removal and better treatment. Sometimes my options are basically ride the highest traffic arterials, fight it out on unplowed miscellaneous crap streets, or not ride. In those cases, I usually will ride the major arterials. It's not a great situation, but I'm a road user too and gotta get from A to B. I wouldn't recommend that to anyone else in a different city without knowing a lot more info about your traffic options... just sharing what works here in Springpatch.

I like my normal bicycle commute route - it is mostly 30 mph city streets with wide lanes and moderate traffic. Snow can really mess with that.

It sounds like the MUP is pretty important to you as a commute route. Upgrading one or both tires to handle the ruts might be a better option for you than getting out in higher traffic or not riding at all. It's a judgment you have to make. Just thought I would share my experience since we are in similar climates.... Good luck, I hope you find a good solution.