Not true, Peter very much wanted Betty in the film. There were certain reasons why a few people didn’t make the jump from the stage cast, and essentially they kept a few more out so the ones they actually blackballed didn’t appear so obviously excluded.
While it's a pity that Andrews didn't do MFL so there would be a record of her performance (no filmed version of the play exists, just excerpts), Andrews herself realized that in a way her not getting the film version allowed her to move on to a major film career she might not have had if she had done MFL. The attitude might have been, "Of course, she was wonderful in MFL, why shouldn't she be. She played it for years on both Broadway and London. But can she do anything else?". In 1964, with the one two punch of Mary Poppins and a non singing straight role in The Americanization Of Emily she proved she was a viable screen presence and cinched it the following year with The Sound Of Music. Harrison on the other hand had a long successful film career prior to MFL (including Cleopatra the year before) so he didn't have to prove anything. While I adore both actresses, I'm more of an Audrey Hepburn fan than a Julie Andrews fan. However, while she's dazzling as the transformed Eliza (why wouldn't she be?), she's not very convincing as the guttersnipe Eliza. We all know she's the elegant Audrey under that disguise and when she emerges from her cockney cocoon, she's the Audrey we know and love.1776 suffers from a real lack of star power. Most of the cast was unknown to the general public at the time.
You can criticize Jack Warner all you like for not casting Andrews instead of Hepburn in MFL but he understood something most don’t—that a big movie like MFL needs star charisma to carry it. As wonderful as Andrews was in the roll on stage as well as in Mary Poppins and subsequent film rolls, in my opinion, she simply never had the screen presence of Hepburn and certainly not the elegance that Beaton and Cukor were aiming for. It makes a difference.
1776 is proof that screen charisma is an absolute necessity to make a big movie memorable, and 1776 the movie just doesn’t have it.
As a son of Cockney parents, I would say the issue is one of being overly precise: there's no comfortability with the dialect. It becomes clear in the scenes with Stanley Holloway, who was Cockney through and through [or rather, "froo and froo"]. There's nothing technically wrong with it, but some readings come off as stiff. Keep in mind that I also say this as someone who loves and appreciates the film for what it is.Audrey is hilarious in the tea scene. Not even Hiller comes close. Also the scene where she first comes to Higgins home. Cukor directed them beautifully. I've never understood the complaints about her cockney scenes.