I totally agree on the whole "Starfleet isn't the Military" issue. Overtly militant fan designs are a huge pet peeve of mine! Can't stand the bloody things and don't get me started on "Starfleet Marines"!.
So who does the Federation call when war is declared upon them? (If there's an answer to this that isn't "Starfleet" then that's really cool.)
The Security Corps.
Starfleet seems to be a descendant of Navy, Coast Guard and NOAA. I tend to think of the "Navy" part that we've seen as being largely defensive
and incorporated within the NOAA/ Coast Guard part. In fact, I'd go so far as to say its overarching historical dominance led to it providing the structure and tradition to Starfleet, even though many of those traditions were a part of the Coast Guard and NOAA as well.
, I think a believable scenario for the hostile environment portrayed in TOS
would lead to there being, in reserve, some serious force that we never saw. Purely military and largely uncrewed, serious -- very serious -- flying weaponry. The kind of thing that would clearly reveal just how much the Fleet had learned from encounters with aggressors like the Doomsday Machine, Nomad or V'ger. The culture would prevent this force from being activated until needed, but fear would demand it be there, in waiting, just in case.