The guys/gals that can express themselves without anxiety are, more generally, going to make a better impression. Those with social anxiety or those with less social skills than those "extroverts" will, more generally, make worse impressions.

Any interview/test/discussion without overwhelming objectivity will be subjective by nature and therefore subject to emotional bias.

People with good social skills are more valuable to employers, other things equal.

The parent mentioned favoring extroverts so I assume they meant hindering introverts, not people with anxiety.

I think they meant favoring extroverts, regardless of what prevents an extrovert-style presentation.