I remember when that movie came out in 1983, it was supposed to propel John Glenn to the Democratic nomination as he was seen as a candidate who could give Reagan a run for the money in 1984. However, Glenn's candidacy went nowhere and if anything, "The Right Stuff" tied right in with President Reagan's theme of "Morning in America" and American exceptionalism. In my opinion, Glenn was portrayed in the movie as a tad bit too squeaky clean, self-righteous and perhaps wrapped a bit too tight.
The movie ran a bit too long as well. If they had cut out the meaningless (and stupid) scenes involving Lyndon Johnson, it would have been a perfect film from beginning to end.
I happened to be an engineer working at Johnson Space Center when this movie came out. I saw in in an audience that was, quite obviously, filled with NASA folks. It was quite a memoriable night.
That was also my impression of The Right Stuff. Glenn was squeaky clean, while the other astronauts had serious character flaws.
I also didn't like the scene set in 1953 in which the radio is playing a song from 1958. Such errors really annoy me.