I use one winter bike that has two sets of matching wheels. One wheelset set has Schwalbe studded tires and another has a Schwalbe semi-slick Cyclocross tires that Schwalbe rate as 4 stars for winter use. In Chicago I use Studded tires about 20% of the time. Like you, I find that a semi-slick does well on a surface that is mostly clear of snow and ice and with smaller patches of snow and ice. A good semi-slick is useful on a little ice and snow if these patches are treated with enough respect. However, the days when the streets are mostly icy, a studded tire is the only safe option. Deep snow is really not ride-able on anything less than a fat bike, IMO.

You might consider getting a fully-rigid vintage mountainbike and putting some fatter studded tires on it for days with plenty of ice on the roads. You will still use your Cross bike with the Conti winter tires more.