I'm thinking for trivia (light or serious), someone with a news background works well, as they're usually well-versed on current and past events. There are the airheaded ones, but that's honestly not as common as sitcoms make you want to think. With a trivia show, when the host breaks down the answer, I want him to sound like he really knows what he's talking about, i.e. Trebek, and not robotic just reading facts off of the teleprompter, i.e. Treadway.
And when I say news, that includes weather and sports. In many cases, a good anchor/meteorologist should be able to ad-lib in the event that their teleprompter goes kaput. I'd be willing to guess that a lot of former meteorologists do so well as hosts because a lot of their job is ad-libbing with the forecast.
Two I always thought could work were Keith Olbermann (probably a tougher trivia game...would love to see him do a week of Millionaire) or Dan Patrick. Another one I thought could work is Bill Weir...he's done some stuff for Good Morning America, and I think he still anchors their weekend version. I second NPH as well.
And yeah, format means everything. For example, would you put Keith Olbermann as host on Minute to Win It?