When a hiring manager specifically excludes overqualified people, this is a pretty big red flag to me. Managers usually only say this when they have a very tedious or low paying job that they think a PhD-type will quickly abandon for something better. I wouldn't exactly call this a "closed door", I'd say it's more of a sign that there are lots of better opportunities for qualified people. If these other opportunities went away, then the tedious jobs would start raising their qualifications too.
PhDs also have many other "open doors" that other people just don't have. All those teaching and research jobs you mention are available to them.