I love Poe, but whether he can be credited with inventing these genres is highly debatable.
On the origin of the mystery tale, Poe published "The Murders in the Rue Morgue" in 1841, and "The Gold-Bug" in 1843.
In 1819 ETA Hoffmann wrote "Mademoiselle de Scuderi".
To my knowledge, it was Hoffmann's work, written decades before Poe's, that was the prototypical detective story (featuring a female detective even!).
On the origin of the Gothic, Poe published "The Fall of the House of Usher" in 1839, "The Masque of the Red Death" in 1842, and "Ligeia" in 1838.
By then, the Gothic genre was well under way, with Horace Walpole having published the classic The Castle of Otranto in 1764. Ann Radclife (the mother of the Gothic) had published The Mysteries of Udolpho in 1794 and it was already parodied by Jane Austen's Northanger Abbey in 1817, long before Poe even started writing any of his Gothic fiction.
Science fiction has a very long history, and depending on how you define it could go back at least to the 2nd Century AD, with Lucian's A True Story (featuring travel to outer space, encounters with aliens, and interplanetary warfare), but if you didn't accept that as "true science fiction" there are plenty of other candidates which predate Poe, such as parts of the Arabian Nights (well over a thousand years old by the time Poe started writing), Gulliver's Travels (1726), Voltaire's Micromegas (1752), and of course Mary Shelley's Frankenstein (1818).
But you can certainly become identified with a genre by being famous for it - and Poe arguably was that.
On the singular scale of "influential," I think that maybe popularity matters more than whatever he "invented," but he was indeed very, very popular.
Furthermore, why can't a number of books and writers be similarly yet differently influential without a requirement of ranking them? Can't there be strawberries, blueberries, and raspberries?
It's helpful to have some sort of comparison, even if they're imperfect or highly subjective (as historical observations often are). With a more cynical take, we can perhaps call it analytical clickbait.
This article doesn't do that. It's just a list of Poe's impressive accomplishments.
However, if we want to talk about the most influential, we should also talk about the breath of their impact. I would give the nod to Twain but maybe Steinbeck or Thoreau would be in the running.
Thoughts? Criteria?
So is Poe influential in the way of VU or influential in the way of Bob Dylan.
The article doesn't actually say, and doesn't make much of an argument for him being the most influential (what argument there is, is more hinted at then effectively made) but given the many things he did I suppose the influence is widespread like Dylan and not direct like VU. If so - I believe widespread influence generally requires quality as well - and the article won't argue that as to him being the greatest.
The Guess Who and Frank Zappa were damn important too.
Btw, one of the most important locations for counterculture rock influence was the Hollywood Landmark Motor Hotel (today: The Highland Gardens Hotel). Jim Morrison, Jimi Hendrix, Pink Floyd, Janis Joplin (who lived and died there), and Alice Cooper were all there... oh and Shep Gordon.
“Rock mythology has it that even though they were around only for a few years, everyone who went to a Velvet Underground concert went out and started a band.”
Several other versions of the quote exist there too. They all support the idea of the outsized influence of a small hyper loyal following. This is consistent with the 1000 (or 100) fans idea.
If the 'most influential American writer' means 'the one most important to the US becoming its own country', then the answer is Thomas Paine. His American Crisis papers buttressed the flagging revolutionary movement and made the common folk actually care. Before Paine's propaganda, the majority of colonists saw themselves as British subjects and the fight for independence was only really supported by the landed gentry.
The American Crisis papers made the common folk care, and without them there likely isn't enough popular support to win the war.
Paine > Jefferson as pertains to the American cause (and Paine > Jefferson as person).
Just then Tom Paine, himself
Came running from across the field
Shouting at this lovely girl
And commanding her to yield
And as she was letting go her grip
Up Tom Paine did run
"I'm sorry, sir", he said to me
"I'm sorry for what she's done"