What nobody here seems actually able to refute is the fact that Sher does nothing more radical to the text than what Lerner and the film do to their text. Yes, My Fair Lady is "its own beast" as you say, but if you actually compare the libretto and the script side to side, the only glaring difference is that one has songs and one doesn't. Writers struggled for years to find a way to musicalize Pygmalion until Lerner finally realized that the best way to adapt the show to a musical was to change almost nothing at all.
Regarding playing the ending as a happy reconciliation, the issue I (and so many others) object to is that however emotionally moved Higgins seems in that final scene, "Eliza, where the devil are my slippers" is not an apology. Furthermore, Eliza returns to Higgins without having witnessed his transformation. She didn't see him experience emotional crisis, think "he's learned!" and then go back to him--she demanded respect, walked out on him, regretted doing so, and returned to his home. In order for reconciliation to happen, Higgins should be the initiator. This was why I so liked the 2002 production which framed the final scene as a chance encounter on the street months later--it showed them politely sharing a moment outside of the toxic power dynamic they had created.
I am reminded of a disappointing semi-professional production I saw a few years back after which an elderly woman said to her husband, "I can't believe she went back to him! It's HIM who should go and apologize to HER! Well, I guess they only had the set of his house built, and not hers..."