Crime and punishment - hmm, could be a good title for a book - have always been worthy subjects for stories, especially morality tales.
I would argue that Poe’s giant step forward was that he split the crime story from the morality story to make it a purer intellectual puzzle.
He also had the timing to be writing after the first police forces in the sense that we understand them today had evolved, so the methods and antagonisms in his stories are still identifiable with today.
And then he included tropes that everybody would later build on: the searching for clues, the impossible crime and the locked room, the least likely suspect, codes and ciphers, the case ripped from the headlines, the adoring assistant to the brilliant amateur who is superior to the bureaucratic forces and can solve crimes from his armchair, and more.
Later writers did not plunder Daniel and Susannah for tropes, they went straight to Poe and adapted (sometimes stole) his concepts and mannerisms wholesale. Doyle certainly did.
That’s enough for me to anoint Poe as the true progenitor of the genre mystery. Doyle added a true series character, whose life could be entered into by fans as adoring as his Watson was, and added a grand sense of atmosphere, but I’ve been trying to think of any innovations that can be laid at his door and none come to mind. I don’t think that even the supervillain almost equal to the superdetective was his. But I stop thinking at midnight and it’s after that now.
Doyle was the popularizer, took what was already becoming commonplace, and had the good fortune to launch a great character when an entire publishing world clamored for mystery short stories.
But Poe created everything and put stories out that lasted despite being tossed into a publishing world that had no concept what to do with them. I see no contest here. I go Poe. (to paraphrase Pogo)