As for John Adams' post-presidential theological musings, they're not mainstream, they're sophomoric and asinine. I haven't even bothered to refute his clippings of a quote here and a paragraph there because of their lack of intellectual rigor.
When he writes [to Thomas Jefferson, October 4, 1813],
"θεμις was the goddess of honesty, justice, decency, and right; the wife of Jove, another name for Juno. She presided over all oracles, deliberations, and councils. She commanded all mortals to pray to Jupiter for all lawful benefits and blessings. Now, is not this (so far forth) the essence of Christian devotion?"
I think, no, it's not the "essence," rigidly biased ideological reductionism. For one, the Greco-Roman vision of the afterlife as the dull gray Hades has nothing to do with the Christian heaven or the beatific vision. I could go on, but Adams is irrelevant anyway, and often laughable. His theory about the religious wisdom of the ancients being destroyed by some churchly cabal is the stuff of cranks, not mainstream Founding religious thought.
I might agree that Adams' idiosyncratic theological musings were his own (especially the conspiracy theory stuff that sounds right out of The DaVinci Code as is Adams' denial of Jesus' divinity). However, the affinity for pagan-Greco-Romanism was quite mainstream among the Founding Fathers. The Ancients were, after all, the progenitors of democratic-republican government, and notable Stoic figures from republican Rome were their heroes.
Indeed, in 1787, in a publicly published book, Adams speaks of a set of laws [Zaleucus'] supposedly revealed by Athena 600 years before Christ as containing sound "religion" that was “rational, intelligible, and eternal, for the real happiness of man in society, and throughout his duration.” Later in his private letters Adams termed Zaleucus' laws to be "Christian."
Van Dyke mentions something about the Greco-Roman afterlife as a dull gray Hades. Well, as historian Peter Henriques shows, George Washington, who out of all the key Founders was most imbibed in this noble pagan Greco-Romanism, often spoke as though he were going to that very gray Hades as opposed to the Christian Heaven. As Henriques writes:
While life goes on - in some fashion - the picture Washington paints of it is generally a gloomy one. “The world of spirits” may or may not be a happy place. When Washington speaks of Patsy going to a happier place, he specifically contrasts it with “the afflicted Path she hitherto has trod.”
A relative had written Washington that his Mother was in fact in a happier place. Washington significantly adds his hope that this is true rather than simply agreeing with the statement. The passing reference to Elysium may well have been made tongue in cheek. While there are clear references to an afterlife and some of them are quite positive, Washington’s references to death and what follows afterwards are more often rather gloomy and pessimistic.
Death was “the grim King” whom Washington, not yet thirty and very near his “last gasp”, feared would master his “utmost efforts” and cause him to “sink in spite of a noble struggle.”
Much later, to demonstrate how much he did not want to take on yet another new responsibility, Washington told Alexander Hamilton that he would leave his peaceful abode [Mount Vernon] with as much reluctance as he would go to the tomb of his ancestors. When people die, he speaks of them as “poor Greene” or “poor Laurens” or “poor Mr. Custis.” Referring to death, Washington wrote about his “approaching decay”, “hour of my dissolution”, of going “to the shades of darkness”, “to sleep with my fathers”, to “the shades below”, “to the tomb of my ancestors”, “to the dreary mansions of my fathers.” Death was “the country from whence no Traveller returns.” The overall image is not a bright one, certainly not a Christian one.
Washington did sometimes speak of the afterlife in cheerier terms. But, again, his notions are decidedly Greco-Roman more than they are Christian:
Washington at least twice makes reference to going “to the world of spirits.”
He writes Lafayette about searching for “Elysium.” [Elysium or the Elysian fields refers to the happy otherworld for heroes favored by the gods.] When Patsy dies of epilepsy, he writes she has gone “to a happier place.” Following his Mother’s death, he reflects the hope that she is in a “happier place.” Washington can hope that God blesses a group of ministers “here and hereafter.” He makes reference to nurturing the plant of friendship “before they are transplanted to a happier clime.” In a draft written by Timothy Pickering to two Philadelphia churches, Washington looks forward to retirement “which can only be exceeded by the hope of future happiness.” While he is dying, he declares several times, “I am going… I die hard but am not afraid to go.” According to Lear’s letter to his mother on Dec. 16th ,Washington told Lear, “I am just going to change my scene.” The image of “going” implies some kind of continuation of existing. It is apparent that Washington had difficulty accepting or conceiving of the idea of nothingness. He does not believe that a person will simply cease to exist upon his or her death.
Washington also appeals to pagan-Greco-Roman authority for the afterlife in his letter to ANNIS BOUDINOT STOCKTON, August 31, 1788 where he wrote:
But, with Cicero in speaking respecting his belief of the immortality of the Soul, I will say, if I am in a grateful delusion, it is an innocent one, and I am willing to remain under its influence....
The felicitations you offer on the present prospect of our public affairs are highly acceptable to me, and I entreat you to receive a reciprocation from my part. I can never trace the concatenation of causes, which led to these events, without acknowledging the mystery and admiring the goodness of Providence.
To be fair, Washington's noble paganism was more influenced by figures like the Stoic philosopher Seneca than Zeus worship. I am no expert in "Seneca," but from what I have read, his view on death (about which he wrote quite a bit) almost perfectly parallels George Washington's. The question of how compatible this noble-paganism is with Christianity is highly debatable. Wiki -- I know not the most reliable source, but a good place to go for "surface" knowledge, later to be confirmed by real sources -- notes the following which seems telling of this entire inquiry about America's key Founders and just how authentically "Christian" they were:
The early Christian Church was very favorably disposed towards Seneca and his writings, and the church leader Tertullian called him "our Seneca".
Medieval writers and works (such as the Golden Legend, which erroneously has Nero as a witness to his suicide) believed that Seneca had been converted to the Christian faith by Saint Paul, and early humanists regarded his fatal bath as a kind of disguised baptism. However, this seems unlikely as Seneca always professed to be Stoic.
Dante placed Seneca in the First Circle of Hell, or Limbo, a place of perfect natural happiness where good non-Christians like the ancient philosophers had to stay for eternity, due to their lack of the justifying grace (given only by Christ) required to go to heaven.