I think Chuck that the pitot can accumulate some ice - enough to affect the reading - long before any significant amount appears on the lift surfaces. I have seen ASI effects when there was no visible ice on the wing leading edges, and in apparent VMC.
Icing can come in many forms, I have had ice build up so fast I was unable to maintain altitude in less than three minutes from the time it started until I could not maintain, and yes there are times when ice can affect the airspeed indication but it is unlikely youn would be flying a 172 in those conditions anyhow and even if the A/S did quit working you can or still should be able to safely fly it.