I've interviewed people with some pretty severe anxiety who have passed it and some who accepted offers. But that is not to say there wasn't someone weeded out that shouldn't have been.
Although for the people who failed the coding part of it I'd say they definitely behaved more overly confident than anxious.
Edit: with that said I have some experience successfully teaching programming to people with disabilities (in a university setting) which probably makes me a fair amount more qualified than many other higher managers. But again, there are no guarantees.
Growing up I played sports all my life and never had a problem in front of crowds. I suspect this is mainly thanks to the cerebellum acting as a co-processor to the pre-frontal cortex of sorts. In an interview however, there is simply way to much ancillary analysis going on to fully focus on the task at hand.
Did it, or did it weed out people who got so nervous at being asked to perform for an audience on demand that they froze up? I have social anxiety, and while I've been super lucky to get jobs through referrals of other colleagues (who introduced me ahead of time to folks I would be interviewing with), if I had to do this in front of a bunch of strangers judging me, I'd probably appear to be very bad at my job. How I act normally with people whom I know is very different than how I act around people whom I feel are just there to judge me and decide how worthy I am.