That's a misquote - that phrase doesn't appear in the letter. I don't think that idea was present either. Could you clarify which phrase(s) conveyed that to you?
If anything, I think he felt compelled to write what he did (not only in this letter) because he felt a sense of imminent danger -- that if the people individually and collectively failed to rise to the high calling of good moral character, the new republic would not last. Remember, the longevity of the United States was by no means a foregone conclusion at the time -- the US Constitution had been ratified a mere 10 years before he wrote this letter, and Adams himself had just been elected as the second president the year before.
(All of which doesn't make his thoughts inapplicable to our time.)
