There are several aspects that are of importance.
Some have already been mentioned- namely, that this has roots in the radio era, that there are aspects that relate to those regarded as being the game show audience, that the quiz show scandals and the demotion of game shows to daytime both are of relevance, and that Bert Parks (and similar hosts in his era- try to watch or listen to John Reed King) played a major role in shaping the stereotype.
Other points, however, have also played a role. One is the question of exploitation- since the radio era (look at criticism Major Bowes received from the welfare authorities in New York City in the mid-1930s), there have been continued questions about the willingness of certain game shows to exploit contestants, either through their personal suffering (Queen For A Day and the programs of that ilk) or through having a sadistic streak (Truth or Consequences, much of the Chuck Barris catalog), and this has helped with gaining an unsavory reputation.
Another aspect, and one that has been an item of criticism since the radio era, is our old pal, "Mo' Money Syndrome". Game shows that have given away a vast amount of cash and prizes for very little work have long been a target, as they have had the image of using this as a means of essentially buying an audience, and this has tended to be applied broadly to understanding game shows generally.
A final point, and one that relates to the stereotype direct, is the image of the game show host as being inherently phony- that the warmth is all on the surface, and that these folk are less pleasant the minute the cameras stop rolling. The issue here is that, throughout game show history, there have been enough examples of this (no names here, but we all should be able to think of a few) that it has tended to stick to the genre as a whole.
This isn't exhaustive by any means, but these points should help in understanding both why the stereotype came into being and why it has had vitality for so long.
