"Only a virtuous people are capable of freedom. As nations become more corrupt and vicious, they have more need of masters." - Benjamin Franklin
I don't believe that many thinking people, whether religious or not, would disagree with the statements that you cite above that virtue/morality is a necessary underpinning of popular government. However the contention that all of the founding fathers believed that a particular and rather narrowly defined religious belief is essential in order to assure that morality exists among the populace is quite another issue. On the contrary many, if not most, showed an acute awareness of a dark side to much religious orthodoxy and viewed it with some very healthy skepticism.
Consider just a few of many statements to that effect: "Of all the tyrannies that affect mankind, tyranny in religion is the worst." -Thomas Paine
“Every national church or religion has established itself by pretending some special mission from God - as if the way to God was not open to every man alike” - Thomas Paine
“Of all the systems of religion that ever were invented, there is no more derogatory to the Almighty, more unedifying to man, more repugnant to reason, and more contradictory to itself than this thing called Christianity”- Thomas Paine
"I cannot conceive otherwise than that He, the Infinite Father, expects or requires no worship or praise from us, but that He is even infinitely above it." Benjamin Franklin - "Articles of Belief and Acts of Religion", 1728 .
"But it does me no injury for my neighbor to say there are twenty gods or no God. It neither picks my pocket nor breaks my leg." -Thomas Jefferson, Notes on Virginia, 1782
So..... You were saying?