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.
There's also another interpretation, which to me would seem more plausible even with the omitted lines left in about potential sabotage. If Stiles is nursing a grudge against an enemy he's never seen personally but heard about, why should he automatically assume that the Enterprise's scan of the Romulan bridge is accurate? It was established from TNG on that the Romulans tend to be baiters and not direct aggressors. They lay a trap and then see if the other power takes the bait, so they can say they're merely "defending" themselves if a war starts.
Given this, why not infer the following possibility: the Romulans can assume that an enemy might be able to scan their bridge; humans and Vulcans are allies and key members of the Federation, their primary enemy; if their crew therefore looks like Vulcans and they're visually scanned, it could create a potential rift between the two races and undermine the Feds. Not every officer naturally would start accusing their crewmates of treachery, but enough guys like Stiles would still make things difficult in an ideal circumstance. Whatever political bickering is created opens up a hole for the war machine to punch through.