A lot of this ground is covered in The Elusive Shift.
It's important to get the history right, I think. Dave Wesley (a wargamer) pulled the idea of an impartial referee making calls on how things should be adjudicated when players come up with wild ideas from free Kriegsspiel, Prussian wargames of the 19th century. This lead to role-play-elements in their wargames. Over time more and more role-playing was added, which lead to Wesley creating Braustein...which Arneson played. Arneson took the idea into the medieval milieu and created Blackmoor. Gygax played a Blackmoor game with Arneson and got Arneson to agree to a partnership and publication. Arneson wrote up the rules and sent them to Gygax. Gygax took those rules, altered them a lot, then published it as D&D. Gygax's contribution was one of editing, alteration, and publication, not creation.
D&D specifically wouldn't have been the impetus for the RPG industry starting, but Wesley's and Arneson's ideas would have still spread. Again, The Elusive Shift specifically covers a lot of this pre-history and early days of the genre. RPGs seem inevitable. But they certainly would have had a less wargame feel and start. The large play-by-mail groups that pre-date D&D might have spread. There might have been a few people trying to publish their rules. Hell, there were games that basically were RPGs without the character sheet, like Diplomacy (1959), but they required a board and pieces. I still think fantasy would be king if it hit around the same time as it did. Lord of the Rings was too much a juggernaut in the related communities for fantasy to not be the first stop for the genre.