Fair enough. Let me rephrase. There are many ways to achieve peace and unity. You can do it through military strength (whether soft power or hard power), or you can do it through diplomacy. I always viewed Trek as saying we'll get there from the latter method. That's not a judgment of the morals or merits of militaries; it's a judgment of the ideal. Ideally, militaries would never have to engage in armed combat. It just so happens we live in a world in which that is sometimes necessary. And I think Trek's view of the future is one in which that call for armed conflict is so rare that the idea of "military" is transformed, such that their essential purpose becomes exploration because defense/war is so infrequently required. In that sense, I think perhaps I (and others) might be more swayed by the Starfleet is a military argument if we have a totally different definition and conception of what a military is 200 years in the future. If the future's definition of a military is "an organization of exploration, science, and discovery, that seeks out new worlds and new civilizations, and defends the Federation when called upon," then sure, Starfleet is a military in that sense. But by current definitions, roles, and standards, I don't think Starfleet is that narrow in its scope.
EDIT: Actually, now that I think about it, Star Trek Beyond perfectly illustrates this transformation. Edison came from a time when Starfleet was a military, exclusively, defending the Federation against the Xindi and the Romulans. By the time we get to Kirk and the Enterprise, that's not what Starfleet is anymore. And Edison believes that's what Starfleet should be once again -- that strife and struggle are necessary, and that peace with former enemies insults his sacrifices. See also: the quote in my signature line.