Watching The Barkleys of Broadway again for the first time in many years, I came to the same conclusion as before about the film, which happens to be the boring old consensus conclusion: not as good as the RKO Astaire-Rogers films, but an entertaining "reunion" picture.
But another thing that occurred to me, which is either interesting or eerie depending on your point of view, is that this is one of at least four MGM musicals that seem to contain thinly veiled references to Astaire's own career. In Easter Parade, his first film for Arthur Freed, he plays a dancer who is primarily thought of as part of a team; when his partner breaks up with him to hit the "big time" (i.e. Rogers' Oscar-winning non-musical roles), he looks for another partner. In Barkleys, he and Rogers play a famous dancing team, and Rogers wants to give up musicals and do serious drama. In Royal Wedding, he and Jane Powell play a brother-sister team, and she quits the act to marry an Englishman -- obviously based on Astaire and his sister Adele. And finally, The Band Wagon is so undisguisedly about Astaire himself that it even opens with somebody auctioning off his top hat and cane and naming "Swinging Down To Panama" as one of his character's movies (Astaire and Rogers were first teamed in Flying Down To Rio).
How much of this was planned, it's hard to say; Easter Parade wasn't written for Astaire, and Barkleys wasn't originally supposed to have Rogers. It's pretty clear overall, though, that Freed and his writers (Betty Comden and Adolph Green for Barkleys and Band Wagon, Alan Lerner for Royal Wedding) were interested in incorporating some references to Astaire's real-life career into the careers of the characters he played onscreen. One of the unusual things about the Freed musicals was that in a time when Hollywood still tended to consider its old movies disposable and forgettable -- older movies were routinely remade, suppressed, chopped up and even destroyed -- the Freed movies teemed with references to earlier musicals and premature nostalgia for early musical movies. All the tributes to Astaire, though they can come off as a little creepy -- enshrining him as a piece of history while he was still actively working -- are part of that awareness of movie history that helped give the Freed films their richness and variety (because the filmmakers, being aware of earlier musicals, had more sources and styles to borrow from).
Incidentally, while I've always been in the Astaire camp in the Astaire-Kelly wars, one difference that's noticeable is that Astaire almost always played a dancer in his movies. Even in Silk Stockings, where he's playing a movie producer, they feel a need to explain that he's also a dancer and have him do a number on stage in a nightclub. Gene Kelly was somewhat more likely to play characters who were not dancers, not in show business.
Correction: Due to a mistake brought on by being totally wrong, I stated that Easter Parade was Astaire's first film for Freed. In fact, he'd already done at least two movies for Freed (Yolanda and the Thief and Ziegfeld Follies).