My main beef with My Fair Lady is that the score doesn't quite mesh up with the pacing. That is, there are these songs in there that are just there for no real reason. They go into "Wouldn't it be loverly" and everything stops while they do a number that doesn't advance the plot or develop the characters or anything.
This is not necessarily bad, and I don't pretend to know exactly what makes a musical number work well and what makes it fail. But I definitely think My Fair Lady is hit-or-miss at best. "Ascot Gavotte", for example, is a hit--it's funny, and it does a good job of characterizing the upper-crust aristocrats. "Get Me to the Church On Time" is a miss--it's a fine song in a vacuum, but the scene itself is a throwaway bit that doesn't need a song and dance number. It's a memorable piece, true, but it doesn't support the pacing or the tone of the play.
That being said, it's certainly possible to go too far in the other direction and end up with songs that move the plot along very nicely while being forgettable and bland, so maybe it's better to err on one side or the other. In opera, they just sing everything and it seems to work out.
...And another thing, what's the deal with Freddie? What a weirdo.