Typical use of cayenne is to keep rodents out of the feed. Some people think it has antimicrobial/immune benefits... I won't vouch for those and especially not in birds. I doubt there's any science for that. But there are even multiple threads about doing so here on BYC.
I've never heard of someone "sticking it up bird noses" but even if you did it wouldn't be any more irritating than any other powder or dust... birds have no receptor for capsaicin that would make it feel "hot" or painful. It would not have any reddening effect rubbed on the skin, either. (Unlike peppermint extract, which has been advocated by a poultry authority figure to my daughter. We talked it out and agreed that we weren't okay with it.)
You can imagine that the pigments and other chemical compounds in cayenne could be valuable. I know in horses that paprika is now popular as a supplement to enhance coat color in black horses that can otherwise fade in the sun. However, one wonders if you would have enough in any feeding program to make a big difference.
Red skin on the face, wattles, and comb to me seems to be indicating a bird that is in good health and one that is laying well. They tend to get more pink as they go towards a molt. But it doesn't seem to be pigment related.
Feeding definitely is going to have some impact on color. You can see how much darker the egg yolks are if you have birds on fresh grass, just as the most obvious example. The birds can't make pigment out of nothing - they need the chemical precursors to do it.