As I understand it, Romans of the Augustan period regarded the Republic as a failed state

Oh good heavens no! The whole conceit of the Augustan regime was that it was a restoration of the Republic! Augustus made a point of remaining an informal power, only keeping hold of the Tribunician Power permanently. Augustus cultivated a literature that celebrated the Republic - Livy is probably the best example of looking toward the Republic as a salve for the crisis of his times (he probably wrote that in the Triumviral period), but Virgil and Varro also showed an interest in Roman origins and continuity. Velleius Paterculus, writing under Tiberius, still called the state a "res publica" I think unironically (though others disagree).

The first indication of "the end of the republic" as far as I know is Tacitus, who was writing under the Flavians. He was the one that said the Republic died with the army of Cassius and Brutus. That's something like 150 years later. Also he has a political interest - he's afraid of criticizing the Flavians (he says so), so he writes about the previous regime, the Julio-Claudians. He'll go back to the Flavians later, but his first attempt at long form history works as a sort of justifying Flavian rule. It just happened that his account of the Republican past caught on in scholarship.

I'm afraid I can't comment on the American Revolution bit, but I hope that correction about Augustan attitudes towards the Republic is helpful.

I did some looking on the American end, and found some sources and historical precedent. It seems pretty clear the Founders were emulating not the corrupt late republic, but the noble early republic. The memoirs of Major General Lee (Washington's number one) refer to James I "being expelled like Tarquin". Tarquinus Superbus was the last king of Rome, before he got the boot and the republic was founded. Parallels between Tarquin and George III aren't difficult to infer.

Another key figure emulated was Cincinnatus. This Roman is most famous for (according to Livy) assuming dictatorial powers, leading the army successfully against the Aquians and the Sabines, and then promptly resigning his office and returning to his farm.

Washington set himself up as a deliberate parallel to Cincinnatus when he resigned his commission and dismissed his staff in 1783, and the Society of the Cincinnati was set up soon after by the Founders (and headed by Washington) to further this deliberate mirroring of the American Founding Fathers and the civic virtue of the High Roman Republic.