Playing without looking is imperative if you are singing into a mic while playing. One cannot move the lips away from the mic to look at keys or frets.
That's one of the best things I've read here in a while: yeah. That actually never occurred to me, but it should have. So many bands like to brag that "everyone sings," but, it's my misfortune or fortune that I just have the Glenn Gould sort of singing voice. IOW, it's pretty much in key, and relatively precise, but it sounds just terrible. It's entirely my fault, not any kind of handicap: just, zero effort on my part. And it shows.
That said, I do know how to use a microphone, and, yes, one cannot...well, I could not, anyway, maintain the right distance from the microphone while also fumbling around with looking at the physical keyboard, nor any notation.
The microphone is itself, in my view, itself an instrument, even if one uses it in pedestrian ways like making announcements and so forth. You know, there's more or less subtle ways to use the diaphragm, in addition to not being a rube and tapping on the windshield or all that stuff everyone hates.
At best, my compromise is to glance at the keys/hands/feet every few seconds, but it's really a subconscious kind of check. I would say a habit, I suppose.
I might be coming up on the stupid side of things, but from reading this forum for a while, I think there's a bit of a gap of knowledge and language between those who are primarily recital or concert pianists and those who regularly play chamber music or other close-combat kinds of music.