Isn't that what companies would look for, though? They want people who can work under stress. If you had two equal programmers, and one who was stressed and the second who was not, any company would pick the second one without thinking.
Ask a firefighter who rushes into burning buildings for a living to give a public speech in front of a large crowd of people in a non-burning building. Which is more stressful? For the firefighter, it may be the latter. Fighting fires is what they train for, what they're experienced with, whereas giving speeches is not.
A lot of people, no matter the profession—firefighter, programmer, etc.—have a great fear of public speaking. Likewise, a lot of people, no matter the profession, also have a great fear of job interviews, especially the crazy audition-style interviews of programmers. It's simply human nature, with little bearing on job performance in general. We shouldn't be giving auditions to people who aren't stage performers. Personally, I never work with someone watching me—definitely not a judgmental stranger who will fire me for any perceived peccadillo—nor do I ever have some specific narrow time limit on my work in the range of 15 minutes or so.
Not exactly. Interviews are a different kind of stress that is usually a direct result of being observed in an unfamiliar and somewhat high pressure situation. You might have to deal with an outage at work, but typically your employment isn't on the line (less pressure) and you have a great deal more familiarity.
Some hiring practices intentionally try to create a more welcome, less stressful experience because they don't want to bias for interviewing skill (as opposed to fitness for the job).
However, a lot of people don't do well under stress, and in many corporate settings, there's usually no real source of stress (except perhaps artificial stress induced by management). Someone might work much better in a stress free environment than someone else under a stressful environment, but you'd then easily miss these candidates by structuring the interview to be stressful.
A bit like 100m race vs. marathon.
In the end, management (or founders if it's a startup) set the tone and who they end up hiring (thus, who will manage your work). And management is just normal people in the end.