I don't see how it's not him, Whitman (the timing of his work is a strong point in his favor, being so concerned with the civil war and that apparently-neveaending struggle in America's metaphorical soul), or maybe Melville though I think he's the weakest of those three. Most other plausible answers are too late in the country's history to take that particular title, for one thing. Irving and Poe are early enough but I don't think either's even close to the combo of clout, staying-power, influence, quality, and breadth of coverage of, specifically, Americanness that one would expect for this (and I like both of them!).
I would agree with other posters that if we allow the search to stretch into the 20th century, odds are we'd end up with a filmmaker or songwriter, but I think we've got our all-around best options already, and they're all (basically) 19th century.
Which of the proposed answers in the linked article would you characterize as one or both of "inaccessible" and "politically charged"?
I think a couple are bad answers (Wilder? She's fine-ish, but America's Homer? Very, very, no) but can't figure out which ones might be striking you that way.
I would agree with other posters that if we allow the search to stretch into the 20th century, odds are we'd end up with a filmmaker or songwriter, but I think we've got our all-around best options already, and they're all (basically) 19th century.