There's no reason that the legislature cannot be the arbiter of constitutionality. This is the case in many other systems, including many Common Law nations with a written constitution but which adhere to the doctrine of Parliamentary Supremacy. At the time of the Revolutionary War there were a diversity of opinions on this, though I think resolving the issue wasn't at the forefront of the debate as other questions about the structure of government (especially Federalism) were considered much more interesting and important.

In fact, the Supreme Court has carved out an exception to judicial review that leave some questions of constitutionality to Congress: https://en.wikipedia.org/wiki/Political_question This is especially relevant given the buzz about impeachment because one would normally think that the definition of "High Crimes and Misdemeanors" is precisely the kind of question our courts were intended to answer.

The U.S. is rather unique in investing constitutional review so thoroughly with the judiciary. Most nations, I believe, either adhere to Parliamentary Supremacy or have a special Constitutional Court even though almost every political system subsequent to the U.S. has a written constitution. Also, U.S. judges have lifetime appointments, whereas most other systems permit the legislature to more easily replace judges (see, e.g. the recent controversy in Poland regarding its Constitutional Tribunal).

The U.S. inherited a very strong normative legal system from Great Britain. The nascent U.S. didn't experience the same turmoil and radicalism as did the French, even though there was a ton of cross-pollination of ideas. Because we had such a strong legal system from the outset I think there was less pressure to put all our eggs into the basket of populist democracy, and were able to leave dormant some difficult questions, both at the state and Federal levels. A few decades later (and culminating in the Civil War) we resolved many of them differently (at least in their technical operation) than Europe did because we had set on our own path so early.

Justice Scalia is often held in esteem for his flavor of Originalism and statutory interpretation, but if you read Scalia's scholarly works he very much believed that U.S. courts (including the Supreme Court) should adhere more closely to a Continental European model, one which circumscribes judicial review to the narrowest possible scope. But that's a normative political philosophy that emerged after the Founding of the U.S. (not to mention on a different continent), which means its absolutely not original in any sense. The 19th century Continental European model clearly emphasizes the supremacy of the legislature as the preeminent democratic institution. But this was not the case in the U.S. Issues of Federalism were of primary concern, and those issues meant that the U.S. was grappling with a more complex definition of "democracy", one that wasn't just about how to allow the population to exercise political autonomy in accordance with moral law[1], but how to allow several distinct populations to exercise and moderate their autonomy in tandem.

[1] Moral law meaning concepts of justice and liberty shaped by Western Christianity (including Greek philosophy and Roman law), the Enlightenment, and the Scientific Revolution.