The Constitution plays an outsized role in the U.S. because our voting populace is quite a bit more conservative than our intellectual elite. Our veneration for the Constitution is why, e.g. bans on interracial marriage were eliminated in 1967 with a Supreme Court case instead of in the 1990s when public support for interracial marriage crossed the 50% line. It's why abortion isn't illegal in roughly one third of the states. It's why schools were desegregated by Supreme Court fiat a decade before the Civil Rights Act was passed.

It's a writ small version of the situation that you see with, e.g. Bangladesh. The voting populace there is very conservative, and despite the constitution creating a secular republic, voters have Islamized the country since then. A few years ago, there was a touchy situation where there was concern an interim military caretaker government would not hand back power to the civilian government once elections were conducted. People like my parents (who immigrated to the U.S. from there) who think secular democracy is really important, kind of supported an undemocratic military intervention that might have the effect of protecting it.