I don't have an exact answer, but I've seen several references in the past to Snow Removal processes that ESRI supports. It appears that the various solutions employ more than just Network Analyst. Here's some of them:
Solutions of Smart Snow Fighting
Managing GIS Operations for Snow Removal for the City of Columbus—Warrior Style | ArcNews
SnowCOP | ArcGIS for Local Government
There is also an older thread on this:
Snow Plow Routing
One specific issue I can think of that will come up with using purely Network Analyst is that the routing process doesn't have a restriction to limit duplication, i.e. the default routing process assumes all vehicles that can be routed can use all roads, so it will have multiple vehicles travel across (plow) the same roads. There are probably ways to get around this, but "out of the box" it has some limitations, so other software/coding/add-ons will be needed to overcome this.
Chris Donohue, GISP