I think the negative stigma associated with calling Starfleet "military" is the head scratching moment for me. Maybe it doesn't fit a contemporary definition of "military" but Starfleet, as a combined service (Kirk's words) allows for both exploration and military operations, at the same time even.I think we just view this show in a fundamentally different way. I see Star Trek as a show about exploration and discovery, not military operations. And in almost every episode, the inciting incident or the foundation of the events that follow have something to do with exploring the universe or meeting new civilizations. I see how people can come to the conclusion that Starfleet is military or quasi-military, even if I have mixed ideas about that, but saying Trek is not about exploration, to me, is just confusing. I think we're watching entirely different shows or something. Maybe this show is like a Rorschach test. I don't know.
And quite frankly we're not even discussing something that matters that much. In Wok, it doesn't matter if Starfleet is a military or not. At all. What matters is the story about revenge and friendship and loss and death. So all this hand-wringing about Starfleet's nature feels a little meaningless at this point. I just see it completely differently.
Is Star Trek about exploration? Sure, but often times there is a military facet as well, and the changes the changes go through in exploring themselves.