The writer seems to be saying MARY POPPINS was a Broadway musical before it was a movie, and of course it wasn't.
Pauline Kael hated HELLO, DOLLY on stage and screen. What she liked was Streisand.
Kelly does obviously reference the movie MEET ME IN ST. LOUIS in DOLLY. He tries, like Cukor tried with MY FAIR LADY, to be Minnelli-esque. They both failed.
"Put On Your Sunday Clothes" does want to be "On The Atchison, Topeka and the Santa Fe." But the latter was inspired by "The Surrey With The Fringe On Top." All those Americana musicals wanted to be the next OKLAHOMA!