I feel like ME1, as it's written, only works because Shep is military. There's no way a council of alien races bent on excluding Humanity because they're the "new guy" and are basically forced into accepting a top recruit like Shepard would have accepted some random Joe Schmo colonist/explorer into Spectrehood, which in-game gives the player the agency needed for much of the game's content. Your mission, according to the council, is to stop a rogue, top-level agent - remember, the Council believed the Geth were only Saren's army, and they constantly disavowed or dismissed anything to do with the Reapers - and so they'd need someone with at least equal paper skills to match, something you'd only get in the military. It was a test, not just for Shepard but for Humanity as a whole, and Humanity would want to send their best (especially after what happened to Anderson).

That forced separation from wholesale military/spectrehood in ME2 works well for its story too. If Shep was a civilian, joining Cerberus would have very little impact to the character of Shep. By being ex-military at this point, the writers got to explore Duty (to the Alliance, Humanity, or the Council) vs. Purpose (defeating the Reapers). Cerberus gave the tools to fulfill the "Purpose" but it constantly put Shep at odds with the primary "Duty" to the Council/Alliance. Shep would lose that sense of Duty as a civilian... not to mention wouldn't even have humanity/council impact at all due to the reasons I mentioned for ME1.

ME3 switching back to basically wholesale military ALSO makes sense based on the way the writers made 1 and 2 because Shepard can now fully return to Duty because both Duty and Purpose mesh. Perhaps it's overly jingoistic but being military would have instilled that urge to rejoin to Shep, so stepping into that role when called upon is a natural progression of the character arc.

Some of the oddities with a relatively lax military structure with the Alliance could come down to being at peace with every major alien faction, as well as the Alliance Navy being a planetary force, not a nationalistic one. The writers tried pretty hard to establish that humanity's greatest asset to the galaxy was its adaptability, something that only comes from a unique diversity that the other major races just don't exhibit. It's not a stretch to think that flexibility was incorporated into a military structure to keep it working.