What you say Steve is true for regular (even wide) tyres. I had already had a crash like that -- precisely as you described it -- instant and face down. That's why I changed to Ice Spiker Pro. The studded tyres I'm using now just
love having ice beneath them. It is because they get full traction just on ice. (Full suspension helps as well, as it ensures both wheels maintaining contact with the ground at all times).
I'm just back from a short grocery shopping at -14 C (wind chill factor -23 C). As Randall and PD said before, the properties of snow are much variable. My Monster rode on frozen packed snow just fantastically this morning! I actually was choosing ice patches for riding wherever they have been. I could watch front wheels of a car slipping at an intersection; Monster crossed the same junction just majestically