I think singers closing their eyes seems to be something where there's a different effect, depending on which side of the Atlantic you are.
Where it's not a comic song, I think it's probably more common than not for the singer to close their eyes back in the Old World, in folk music circles that is. If anything it's when a singer keeps them open and looks around that it feels affected and artificial, which seems to be the other way round to how Americans see it.
In fact I think that's one reason why, when I'm singing a song from a written copy, it never really seems to feel right.
So I suppose that means I'm playing blind too - never thought of that. If it's a tricky bit, I suppose I'd be looking to see what was happening. I'll have to check.