No he doesn't "shoot with a Leica." As he states in the Instagram discussion quoted above, he used the iPhone for many of the photos in his prize-winning book,
From a True Story. The Fuji GFX medium-format digital that he's shooting with right now is a camera he uses only occasionally. Currently, he's shooting mainly with the Fuji XT2. I think he still has his M6 that he likes but shoots with rarely. In addition to the iPhone, last year or so he also occasionally shot with an Olympus underwater point-and-shoot camera, the TG5 (?).
I like his camera-agnostic attitude and, as I said, his openness to new technology. He is very approachable and does a lot of free mentoring as well. It's easy to question peoples' motives, but he doesn't have a conflict of interest simply because he isn't the sort of guy you're suggesting — it's easy to question the motives of people you don't know. Also, as you can read above in the last statement, DAH says he's looked at the pictures of the person he is writing to and gives his judgment of what that persons needs.
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Alone in Bangkok essay on BURN Magazine