Yeah, I think I've thought about Let It Be solo, and I don't think it would work for one reason. The guitar solo would be cut out. And that's a huge part of the song. If Paul played it solo, the song would seem empty during that part where the solo should have been. But that's just me.
I seem to remember Paul playing a piano version of Let It Be at the Live Aid concert in 1985. Apart from problems with his vocal microphone, it seemed to go down really well.