I may very well be wrong but I have never thought that the book Higgins’s Universal Alphabet was primarily concerned with putting forth a method of recording "visible speech." (I just noticed that Shaw has it as Higgins’s. Did Lerner decide that Shaw was wrong when he changed it to Higgins’? Or was it changed by the person who copy-edited or proofread the published script?) I’m glad that you brought this up as you led me to go searching for more info.
Higgins says to Pickering just before Eliza enters in Act II that they will take her down first in Bell’s visible speech and then in Broad Romic. Henry Sweet, the inspiration for Higgins (for those who may not know or may have forgotten), invented the latter in the 1870s while Bell’s visible speech was invented in the 1860s. Broad Romic was Sweet’s attempt to build upon and improve Melville Bell’s method of notating pronunciations. (I imagine you know all that. I hope that I’m understanding it correctly.) If Higgins’s Universal Alphabet was meant to be understood as a method of recording speech, wouldn’t Higgins say that they will take her down in that?