Even though the Garden of Eden is alluded to in just one line of "Nothing Gold Can Stay," this mention opens up the poem's meaning tenfold. It's the human complexities of this Garden, and not the botanical ones, that make this one line of the poem capable of adding much more meaning to the poem as a whole. When the speaker says "nothing gold can stay," he's not just talking about the blooms of a willow tree, but about human innocence and joy.
The allusion to the Garden of Eden in this poem makes its meaning more universal—it's less about nature and more about humans everywhere.
The reference to the Garden of Eden solidifies this as a nature poem, through and through. The speaker is lamenting the fact that we were banished out of natural paradise thanks to original sin.