I ride my cross check year round with a set of studded tires and in IGH in the winter. It's my commuter/town bike so I have a set up for both seasons. I also have a Moonlander now so I'll be rocking that in this winter too. All of my bikes are good bikes, and I'm not afraid to take them out. I take care of them and they work great.

I don't know where you live or how severe your winter weather can be, but I would be cautious about running a "cheap" bike in the winter with suspicious parts, only because it's the worst time to deal with mechanical failures out on the road for obvious reasons. IMHO it's preferable in all seasons to have reliable equipment. That doesn't have to mean expensive, but checked, solid, and well cared for to avoid winter breakdowns. YMMV of course.