From what I know, he was well trained from early childhood, had talent to boot, and lucky enough to end up composing on the court. Being a workaholic helped too.
I'm almost refraining from making a judgement, but if I had to force it - no, he's no genius in my book. Once again, I really could be right as well as wrong about that. That doesn't prevent me from having a pet hypothesis that him, being labeled as prodigy right off the bat, got him into a situation where he was allowed to compose tunes that would appeal to many, as opposed to your generic court composer, who bows to kings wishes. Which is a sort of little revolution right there - court approved pop music. For all I know, he's the 18th century's Lady Gaga.
But not to stop your point right there, I can say that Frank Zappa would qualify as genius for me. I can really follow his music and feel his firm grasp of rhythmic patterns along different sections as well as his control of general timbre. He even called his process of music composing/performing as "sculpting the air". I can sense him creating a quite literal model of reality right there. And, unlike for Mozart, I kind of have much more information about the current state of the matter when Zappa came in.
Amongst other musicians I consider genius - Iannis Xenakis, Arnold Schoenberg, John Cage, Edgard Varèse... You'll find them usually labeled as experimental or avant-garde. Many quacks in that genre though.