This had its good points and bad points; the Big Ballet soon became a cliche of musicals, and prompted one critic to write "Cripes, what I wouldn't give for a good old hoofing chorus!" But through his Broadway work, Balanchine expanded the scope of what dance could do in a musical, and made it possible to view dance not as a diversion, but as part of the story. And by bringing his own highly distinctive style to the shows he choreographed, Balanchine established the idea that a Broadway musical could have choreography just as distinctive and creative as the ballet, and borrow from "serious" dance without losing its showbiz savvy (On Your Toes even makes this into the basis of the story: a hoofer, played by Ray Bolger, and a ballet dancer, Tamara Geva, pool their talents in a fusion of ballet and showbiz, classical and jazz, "Slaughter on Tenth Avenue"). It's Balanchine who paved the way for the Broadway careers of such choreographers such as Agnes De Mille and Jerome Robbins.
One thing I noticed, when typing those names, is that while many choreographers were able to cross over between ballet and Broadway, few of those choreographers did much work in Hollywood. The odd thing about that is that Hollywood musicals aren't less dependent on dance than Broadway musicals; if anything, they're more dependent on dance. The greatest Hollywood musicals tend to be dance-heavy, and the two biggest stars of movie musicals, Fred Astaire and Gene Kelly, both gave up Broadway stardom for Hollywood. The big Broadway stars, with occasional exceptions (Ray Bolger, Gwen Verdon), were usually singers (Al Jolson, Eddie Cantor, Ethel Merman, Mary Martin) or comedians who could sing better than they could dance (Bert Lahr, Phil Silvers). Yet there aren't many famous movie choreographers; under the influence of Gene Kelly, MGM did a few Big Ballets in the early '50s, but mostly the dancing in movie musicals is an expression of the dancer's personality, rather than the choreographer's; Fred Astaire may have worked out the steps with Hermes Pan, but it's still a Fred Astaire dance, whereas "Slaughter on Tenth Avenue" is unmistakably Balanchine's dance, not Ray Bolger's -- the proof being that people can, and do, perform "Slaughter on Tenth Avenue" without Bolger; you can't do the "Cheek to Cheek" dance without Fred and Ginger.
I suppose I've just answered my own question about why choreography never took off in movie musicals: stage musicals need to have dances that can be performed by someone other than the star, so the choreographer's vision becomes more important; and as shows started to run longer, became still more important (as musicals needed to run longer to make back their money, it became more important to have choreography that could hold up under multiple cast changes). This doesn't apply with movie musicals, where everything happens just once. There was one movie choreographer who became famous -- Busby Berkeley -- but Berkeley had not particular distinction purely as a choreographer; in his Broadway work, he was a pure "dance director," specializing in unison dances with the boys and girls repeating the same step over and over, and he carried that style over into his movie work; he became famous, and deservedly so, for what he did with the camera, not for what the dancers were doing.
Another point in this rambling post: I get the impression that by the '50s, there was a bit of a backlash against "arty" choreography. Maybe 1951, with the double-punch of Gene Kelly's interminable Big Ballet in An American in Paris and Jerome Robbins' very long "The Small House of Uncle Thomas" ballet in the year's big hit musical, The King and I, was the moment when people started thinking that these big dances were eating up too much time. In any case, the '50s brought fewer and fewer ballets in musicals, more old-fashioned showbiz dancing, more choreographers like Bob Fosse who cultivated non-balletic styles, more musicals like My Fair Lady with relatively little dancing (even Robbins, who created the ultimate dance musical with West Side Story, followed it up with Gypsy). The backlash was even written into a couple of movie musicals by old hands who had lived through the dance revolution; Irving Berlin, in a not-particularly-good song for his not-particularly-good movie musical White Christmas, lamented, "Instead of dance, it's choreography," while Cole Porter added some new lines to the movie version of Silk Stockings:
It's not enough today to see a dancer at his ease --
He's got to throw his back out and come sliding on his knees.
He's got to have glorious Russian ballet or Chinese ballet or Hindu ballet or Bali ballet or any ballet
And stereophonic sound.
Finally -- and forgive me for writing such a disorganized post, but chalk it up to post-Yom-Kippur disorientation -- the booklet for the cast album of House of Flowers, a 1954 show (written by Truman Capote, with music by Harold Arlen), has an anecdote about Balanchine, who was originally hired as the choreographer but -- like the director, Peter Brook -- wasn't the right choice for a show about rival Haitian brothels. Geoffrey Holder, one of the lead dancers in the show, recalled that Balanchine's approach was a bit over-formal:
We waited two days for Mr. Balanchine to teach us the mambo. All he had to say was "Mambo!" and we would have done it because we did it every night at the Palladium. But all of Balanchine's ballets are mathematical. So it took him two days to break it down.
Balanchine quit or was fired during tryouts, and was replaced by a younger choreographer, Herbert Ross (who also took over the direction when Brook was fired, making House of Flowers a possible contender for the record of "most distinguished list of creative staff who bombed out and got fired").