The thing is that doesn't even sound wrong and I don't think any native speaker would object to the article when used with a place of origin/belonging in a speech.

It emphasizes the statement if you ask me.

Only many years after learning of the speech I read online that it's allegedly "a gaffe".

In this specific case the "ein" emphasizes/expresses that he "is one of us" to the Berliners. Not a gaffe at all, in my opinion.

To me, "Ich bin Berliner" sounds relatively neutral, or like a short statement. "Ich bin ein Berliner" sounds a bit less direct or a bit softer to me because you state that you are just one of the citizens of Berlin. But this may be just my feeling.

(In case you put the emphasis on "ein", you would clearly emphasize that you are just one of the many Berlin citizens.)