The house itself is a modification of the Greek Revival style whose popularity had drifted across the pond from the likes of James 'Athenian' Stuart but whose classical allusions were uniquely appropriated in America as a noble style for a still-young democracy.
Its construction, being wood when a similarly sized English house would certainly be of stone, its brick foundation, and most especially its decorative features--the shutters, the picket fence, and the wonderful added fillip of trellis--are quintessentially American.
Greek Revival houses are highly symmetrical, and so have a strong natural axis running through them; the line on which you could place a mirror and both sides would look alike. It is so logical for this to extend from the house into the garden that a Greek Revival (or any strongly symmetrical) house looks lost in its landscape without that continuation of the axis. It would be more common for this to be done using a straight allee of trees; here the axis continues through a circular drive which would have been a turn-around for carriages. It is perhaps a reflection on Georgia's mild climate that the turnaround is not nearer the house, as would be more typical. At Bellwood, ladies would rarely need to worry about disembarking in the rain.