Yes, the US is a republic in the traditional sense, because in the traditional sense, a Republic is simply a government without a monarchy.
Even in the more "modern" sense of the word, the idea of a representative government, the US is a republic, because unlike, say, France, there are no plebiscites or referendums at the federal level.
Hmm, here's an idea - if the US chose the model of a parliamentary democracy, could it be similar to modern Austria in OTL?
Actually, no, the traditional sense is just a government by rule of law. Early Modern republicans, like Locke and Rousseau, were completely fine with the idea of a monarchy. So long as said monarchy reigned by rule of law.Yes, the US is a republic in the traditional sense, because in the traditional sense, a Republic is simply a government without a monarchy.