There are also suspicious aspects to the timing. Spock is supposedly a rare example of a Vulcan serving among humans in Starfleet. Now, after a century of supposed inactivity, Romulans make their move - right when this rare starship with the rare Vulcan assumes patrol duty, possibly for the very first time in her service history since Kirk sees the need to have Spock give the primer speech...
These are things that might go "click" in the mind of a person who already sees spies everywhere - but also a person who has just been shocked into believing in spies for the first time.
Also, the idea of the totally faceless war is cool and fantastic and all... But difficult to execute in practice. There might well have been rumors during and after the war, rumors that only diehard hatemongers like Stiles would still remember.