I'd love to see it. Not necessarily a majority of the time, but more of a variety. To use a cooking analogy, the Les Paul is like beef and the Strat is like pork. Les Pauls have a richer, thicker tone and are better as a stand-alone entree with just salt/pepper. Strats have a thinner tone, and are a better pallet for different flavors and seasonings (ie: effects).
The sound of Goldie (and whatever effect that is/was on "I know where I belong") was to die for, and likely would not have sounded as good with the same effect used with a Les Paul. http://youtu.be/AzOVoSnchw4
I'm familiar with the circumstances that led to the departure from Fender, and don't blame Joe in the least for his decision at the time - but I think the point has very much been made now. It surely was Fender's loss and Gibson's gain.
I just miss the variety of tones and effects used from song to song in the early days. Joe has developed a signature sound/tone now (and a mighty fine tone it is) but it's fairly consistent from song to song when playing a Les Paul. Maybe there's room for more Strat material in the future?