'Bewitched Bothered and Bewildered' comes from the Rodgers & Hart musical Pal Joey, which in its original 1940 run starred a then unknown Gene Kelly (most famous for the movie musical Singin' In The Rain). Kelly played the title role, an conniving, pushy, nightclub entertainer having an affair with an older, married woman. In Pal Joey, 'Bewitched Bothered and Bewildered' is relatively early in the show, as the older, married woman expresses her feelings about her new affair.

According to a 1978 paper called 'Pal Joey Meets More Than His Match' in the Journal of Popular Culture by John Harrison, Pal Joey was the first American musical with an anti-hero for a lead, and it indeed was controversial. I can't find the original New York Times article by Brooks Atkinson reviewing the show, but one passage is widely quoted in writing about Pal Joey:

...if it is possible to make an entertaining musical comedy out of an odious story, Pal Joey is it. The situation is put tentatively here because the ugly topic that is up for discussion stands between this theatregoer and real enjoyment of a well-staged show...John O'Hara has written a joyless book about a sulky assignation...can you draw sweet water from a foul well?

Elizabeth Jordan in America magazine in 1940 said that:

One can only say that the author presents his unpleasant leading character exactly as he is, that Gene Kelly makes him as unobjectionable as he can be made, and that the audience takes him in, sees through him, rejects his type, and accepts the new comedy.

A seemingly anonymous writer in Time magazine (at least, the academic database I found this on didn't list the author) said that:

For those who can park their morals in the lobby, Pal Joey is a wow.

So yes, the morals of the situation were clearly front and centre to some critics. Others applauded the risk the producers of the musical had taken in portraying more recognisably human characters on the stage. For Louis Kronenberger it was the most "unhackneyed musical he had ever seen", while Walcott Gibbs applauded that the musical had "three-dimensional characters talking and behaving like human beings".

According to Harrison, the 1952 revival of the musical was more successful, and attributes this success to the fact that the 'innuendos and frank lyrics that had shocked audiences and critics in the 1940s now seemed more acceptable and more palatable'.

It does occur to me that the original Broadway production of Pal Joey mostly just predates the 1940s film noir films like The Big Sleep, The Maltese Falcon, and Double Indemnity, where infidelity and imperfect anti-heroes do abound. Like those films Pal Joey was based on a story by a writer from a certain 1930s literary tradition; John O'Hara who wrote the book Pal Joey was, like Raymond Chandler or Dashiel Hammett, known for his ability to look at the grim underbelly of America, though O'Hara was less focused on private detectives. I imagine that, after a decade of films like the 1946 Double Indemnity (which features an insurance salesman contemplating helping a housewife he's besotted with murder her husband), the musical Pal Joey would have seemed a bit less risque than it had in 1940, but I'm not sure if that was the specific reason behind the laxer moral standards that greeted the 1952 revival.