After The Band’s 1969 second s/t album---routinely cited as the first "Americana" album---sent shock waves through the Rock music world, suddenly The Dead---and a lot of other bands---followed suit, but none with the depth The Band possessed and exhibited.
I’m not a deadhead but I enjoy and admire the Band and the Dead’s Americana phase equally. Personally, I don’t perceive the Band’s tunes, overall, displaying a greater "depth" but each to his/her own. I do regard Hunter and Garcia as a significantly underrated songwriting team.
As far as the Dead being first and foremost psychedelic, here is a pertinent quote from the New Directions In Music site:
... Robert Hunter was not just a poet... and lyricist; he was genuinely steeped in traditional American folk music and he and Jerry Garcia were playing the cafe scene together before there even was a Grateful Dead. From Garcia, he learned a lot of traditional songs from the bluegrass and jug band genres and he knew how to structure his lyrics to fit these musical forms.