I wouldn't put the starter in the fridge w/o activity. When it cools, guess what happens. Your fridge is hardly the cleanest place, most likely. With activity in the airlock, it will continue pushing out as the wort cools, making sure you don't suck anything in.
As for what it does for you, you're right about giving you shorter lag time, but the whole point of that is better flavor. When your wort cools, everything wants to get at that sugar and there's some fierce competition going on. The sooner the yeast you choose can take hold of the territory, the sooner you get rid of all the yeasts you didn't choose, as well as any small amount of bacteria that might be around. The time before this happens is the time that off-flavors get into your beer.
Another, smaller issue is that when the yeast are reproducing, before fermentation, they are using your sugars as an energy source. It's debatable how much affect it will have on the palette, but I work hard to get a high efficiency in my mash, and am all for giving the yeast something else to chew on until they're ready to get to work.
That said, I have to confess I've been pretty happy with my results using WLP by itself. But sitting beside me on my desk is my very first starter in a gallon jug, awaiting my porter tomorrow.