"Beautiful Girl" seems to have one purpose and one purpose alone: So Cosmo can notice Kathy among the dancers, and tell Don, re-uniting the two. The costume montage is especially BLAM-ish....
"Beautiful Girl" is actually making fun of the types of musical numbers (and songs) that appeared in musicals back then, pre-42nd Street.
Excuse Plot: The film was specifically written as a setting for songs from MGM's back catalogues.
Franchise Original Sin: After the proliferation of "jukebox musicals" in the 2000s, which contrived their stories around pre-existing songs, people very quickly soured on the idea. This can make it weird to realize that this film, considered one of the greatest musicals ever, was also one of them using songs from the 1920s, even if the term didn't exist at the time.
Meta example: The reaction of the preview audience to sound being added to a movie that wasn't written/designed with it in mind is very similar to modern reactions to films that had 3D added to them when they were not filmed that way.
One-Scene Wonder: Cyd Charisse in the "Broadway Melody" sequence, without even a single word of dialogue. Also Julius Tannen, playing the guy appearing in the short film demonstrating the new sound technology.
Padding: The film is only about an hour and a half long, and quite a lot of that time is taken up by musical numbers that have almost nothing to do with the plot, like "Beautiful Girl" and especially "Broadway Melody". However, those numbers are so awesome that it's difficult to care.
Special Effects Failure: When Don runs into the building that explodes, the characters off to the side change position with the cut.
Uncanny Valley: The frozen tableau of girls in the "Beautiful Girls" number is a little off-putting.
Values Dissonance: Don gives a completely false account of how he got into acting, mentioning having attended prestigious acting schools while he actually honed his acting skills as a street performer and in minor, lackluster roles. This would be an utterly baffling thing to do from the perspective of a modern viewer, who would be much more likely to relate to a classic ‘underdog’ story instead of that of a person who rode his privilege all the way into glory.
Vindicated by History: The film did disappointing box office and was generally snubbed by the Oscars, but is now seen as one of the great movie musicals of the 20th Century.