This debate was not settled, and there was no winner, then or now. In fact, the societal fault lines outlined by the question were shifting even in the Twenties.
Generally, the differences in worldview between so-called "modernists" and "traditionalists" were associated with rural and urban Americans. In the 1920s, advertising, communication, and mass media exposed a wide variety of Americans to what might best be described as an urban culture—one that was remarkably modern. At the same time, "traditionalists" were quick to use the same tools to popularize their worldview, especially evangelical Christians like Billy Sunday and Aimee Semple McPherson. Thus, there is a sense in which "traditional" ideas became "modern" in the 1920s.
Public opinion polls were a new phenomenon in the 1920s, so it is very difficult to measure how Americans actually felt about the issues that were associated with modernity and traditionalism. Mass media and the fact that increasing numbers of Americans moved to urban areas contributed to the perception of a cultural divide, and evangelicals, nativists, and many others responded visibly and vocally.
While conclusions are difficult to reach in terms of who "won" the debate, what we can say is that these divisions persisted. As early as the 1930s, "traditionalists" found common ground with corporate interests, the ultimate modernists, in their opposition to the New Deal, which they characterized as socialistic. This coalition between the so-called "religious Right" and business interests would be a powerful force in conservative politics beginning in the 1960s.