I didn't see Laura in her Tony-winning role, but taking that into account, along with her performance in She Loves Me, Laura is ready for a major original role. She is smart and funny and audacious, and of course she brings that voice along with her.
Eliza is not so meek. She goes to Higgins' house to follow up on his offer. My memory failed me. I just remembered her constant cries of "I'm a good girl, I am." Still, cockney Eliza doesn't bring to mind Laura to me.
Now, to go off topic, speaking of Julie Andrews and My Fair Lady, although it's not one of the major highlights, I've always liked "Show Me." I think that Julie likes it too since she sang it quite often when given a one hour special on TV or was a guest on a musical variety show. (Big stars like Nat King Cole and Judy Garland and Perry Como and even Julie Andrews were given a hour to entertain, with guests. Of those I mentioned, only the show of the least talented, Como, was a lasting hit. Julie lasted a year, if that long. But today we can support 12 essentially similar crime dramas per week.)
Anyway, this 1962 special on the music of Lerner and Lowe brought together the stars of Camelot, My Fair Lady (no Rex Harrison) and Maurice Chevalier together and they recreated some of the original numbers from the shows. For instance Stanley Holiday's two production numbers are here. But here is Julie recreating her role in My Fair Lady. I'm not sure who's playing Freddy.
https://www.youtube.com/watch?v=-R_lv6_5Mvg