Sondheim was criticizing Hart as early as the 1980s, if not before. I specifically remember having conversations about this in the 1980s.
As you wrote earlier, the shows themselves are very popular, but the songs don't really resonate beyond the context of the shows. Even Webber didn't have all that many "hits" in the traditional sense of the word.
Of course, popular culture itself has increasingly fragmented over the decades, and Broadway would be yet one more example. The fact that many of us can't name a single song that Sondheim wrote since 1973, despite his resume, would attest to this.