People without that niggling sense of anxiety always in the background, the folks who are easily going to ace the programming interview because their anxiety levels are so low, those folks, I theorize, are potentially not going to be so careful. Now you could argue that that's a plus in many situations - a startup that needs to get code out the door right away, for example. But it really depends on the domain. For critical systems in applications like healthcare, avionics or robotic control I think the anxious coder is the one you want.
Companies that completely weed out the anxious programmers, as you imply, will have a different culture with perhaps too much emphasis on risky behavior.
Could this be deliberate? As working with people with anxiety problems is often a pain in the ass as they won't report problems for fear of seeming stupid or ask for help.
Confident people are far easier to work on a team with.
Google is basically using anxiety to filter good candidates and eliminate false positives which works in a sense but is still highly illogical.
Why not use a technical filter to filter for technical candidates? Anxiety seems like a pointless filter.... how does that even eliminate false positives?