Even a corrupt Roman Empire very likely would have proved a more stable bulwark against 1300 years or more of Islamic expansionism, it surely would have proved a more secure library of the wisdom and science of antiquity which, as you point out, our founders found instructive.
Rome seems to have been done in by about 150 years of severe cold weather due to a spasm of volcanic eruptions and the bubonic Plague of Justinian, which killed about half the population. I am not at all confident that we could withstand such a combination of calamaties along with Islam on the march.