How common is it for warm fronts to be responsible for record high temperatures?
I'm not sure that record temperatures occur with enough frequency that we can attribute them to this specific phenomena, though I am no authority on records.

I think that attaining a record high with warm frontal passage is unlikely.  As you mention, the warm front is often accompanied by cloudy weather, and these clouds first manifest as high cirrus clouds well poleward of the surface front, lowering closer to the front.  This combined with possible precipitation will moderate the temperatures.  If rain is falling on the cold side of the front prior to frontal passage (FROPA) it will likely stabilize at the wet-bulb and stay far from the record temperatures.  Places on the warm side, if experiencing precipitation will have a similar experience and probably not near high temperatures.  You may have a case here for potentially warmer night temperatures, but I'm not sure you'll be in record territory.

A better case for potential high temperatures can be just prior to cold frontal passage.  Especially for a strong cold front, the pressure gradient can drive strong southerly flow (in the NH) bringing lots of warm air advection ahead of the front.  If you watch a temperature trace before such FROPA you'll see a sharp rise in temperature before it plummets. 

Typical FROPA probably won't set records.  You need some exceptional aspect of it or at least some unseasonable occurance to really threaten the records.  

Though I was unable to justify it with data, I do not suspect warm frontals are a common source of high temperature records (high daily max or high daily low).