Of all the dispositions and habits which lead to political prosperity, religion and morality are indispensable supports. In vain would that man claim the tribute of patriotism, who should labor to subvert these great pillars of human happiness, these firmest props of the duties of men and citizens. The mere politician, equally with the pious man, ought to respect and to cherish them. A volume could not trace all their connections with private and public felicity. Let it simply be asked: Where is the security for property, for reputation, for life, if the sense of religious obligation desert the oaths which are the instruments of investigation in courts of justice ? And let us with caution indulge the supposition that morality can be maintained without religion. Whatever may be conceded to the influence of refined education on minds of peculiar structure, reason and experience both forbid us to expect that national morality can prevail in exclusion of religious principle.
-- George Washington, Farewell Address.
Our Constitution was made only for a moral and religious people. It is wholly inadequate to the government of any other.
-- John Adams, October 11, 1798.
This preamble [to the laws of ZALEUCUS] instead of addressing itself to the ignorance, prejudices, and superstitious fears of savages, for the purpose of binding them to an absurd system of hunger and glory for a family purpose, like the laws of Lycurgus, places religion, morals, and government, upon a basis of philosophy, which is rational, intelligible, and eternal, for the real happiness of man in society, and throughout his duration [My emphasis].
-- John Adams, DEFENCE OF THE CONSTITUTIONS OF THE UNITED STATES, 1787-88.
The laws of ZALEUCUS were supposedly revealed by Athena 600 BC. When Washington et al. stated "religion" was necessary to support republican government, they meant "religion" not "Christianity." Though they weren't familiar with all world religions, they did mention Christianity, Judaism, Islam, Unitarianism, Deism, Hinduism, Native American Spirituality, pagan-Greco-Romanism, and Confucionism as "sound" or valid religions.
The interesting dynamic -- the kernel of truth in the Christian America crowd's argument -- is that many in the masses of that time, especially many orthodox leaders, probably did hear "Christianity," or "Protestant Trinitarian Christianity," when for instance George Washington stated "[a]nd let us with caution indulge the supposition that morality can be maintained without religion." Indeed, the much spread spurious quotation of Washington's, "It is impossible to rightly govern without God and the Bible," is a paraphrase of some pious figure's interpretation of Washington's Farewell Address. The problem is, that's not what Washington said, and it's not what he meant. But Washington did not, for instance, when the pious clergymen wrote him, correct their misimpression by stating, "I know you believe in the Trinity and there is only one way to God, but I believe all religions are valid." That would just cause needless trouble.
So the systematic use of generic and philosophical religious language did well serve the Founders in their need to thread the needle between their heretodoxy and the orthodoxy of the masses (or at least the orthodox ministers who held great positions of institutional social power; arguably the masses were nominal not orthodox Christians). However, it did lead somewhat to "talking past one another" when discussing broad abstract concepts. People often agree on broad general concepts like that of "religion." Who doesn't think we have "rights" to life, liberty, and the pursuit of happiness? But when right to life means things like abortion, or right to liberty means whether to legalize drugs, that's when the disagreements begin. America's Founders needed to unite, not divide. So it would have been unwise for them to publicly pick fights with the orthodox. Thomas Paine did and paid a great cost for it.
Though there is a big difference between "religion" meaning "orthodox Christianity" on the one hand and meaning "religion generally" on the other. Over time, those differences would play themselves out in profound ways. America for instance, became a nation where the law (indeed fundamental rights in the US Constitution!) treated non-Christian religions equally with Christianity because of America's Founding principles.