I think the climate of fear you live in as an airline pilot can lead to accidents.
We were very open with each other, would always let the other know of our alertness. Knowing a friend is in distress and needs you to pull through for them is worth 5 coffees. I'm not sure a passenger pilot would tell his copilot that he had a rough night and is going to battle to keep awake.
Like Dirty Harry used to say "A man has to know his limitations".
People need to know that pilots can fall asleep and procedures should be in place to stop any accidents when it happens, not fight the laws of nature by trying to select super humans as pilots.
As aircraft become more automated this problem is going to get a lot worse. IMO the simplest solution is for the computers to mix it up, data entry and operation is randomized. Altitude is not always the second entry, values have to be entered including their units of measurement. Sound and visual alerts begin with a random sound/light sequence then revert to normal values. The result should be that no two warnings/alerts ever sound/look the same because in real life no two warnings/alerts are the same as they occur at different moments in time. A human brain is hard wired to wake, using a limbic response, if it senses danger, danger is identified as an unusual or a known dangerous sound.