why shouldn’t they promote their state of education to relay their level of comprehension and intellect?

People with PhDs generally have a deep understanding of a narrow subject. And outside of that narrow scope, some of them do things like try to fill a bucket with more water by extending it at the top with cardboard. And that type are the most likely to compensate for their other inadequacies by relying on their academic achievement.

In my experience the people with PhDs whose intellects I have the most respect for are the ones I’m least likely to hear about their doctorate from. It’s not a marker for comprehension and intellect, it’s a marker for obsessive perseverance and interest in an academic field, plus an ability to write technical language in that field.

People who say unprompted that they have a PhD as proof of their intelligence probably don’t, and most likely don’t spend much time dealing with PhDs either.