The answer is a definite yes, in particular for the example you have given. The desecration within Athens may have been intended to play upon the superstitions of normal Athenians, but amongst the aristocracy and the city's rulers, this was immediately seen as a political act intent upon sabotaging the Sicilian Expedition. Following the desecration of the statues, there were accusations levied against Alcibiades and his followers, as detailed by Thucydides:

27In the midst of these preparations all the stone Hermae in the city of Athens, that is to say the customary square figures, so common in the doorways of private houses and temples, had in one night most of them their faces mutilated. No one knew who had done it, but large public rewards were offered to find the authors; and it was further voted that any one who knew of any other act of impiety having been committed should come and give information without fear of consequences, whether he were citizen, alien, or slave. The matter was taken up the more seriously, as it was thought to be ominous for the expedition, and part of a conspiracy to bring about a revolution and to upset the democracy.

28Information was given accordingly by some resident aliens and body servants, not about the Hermae but about some previous mutilations of other images perpetrated by young men in a drunken frolic, and of mock celebrations of the mysteries, averred to take place in private houses. Alcibiades being implicated in this charge, it was taken hold of by those who could least endure him, because he stood in the way of their obtaining the undisturbed direction of the people, and who thought that if he were once removed the first place would be theirs. These accordingly magnified the matter and loudly proclaimed that the affair of the mysteries and the mutilation of the Hermae were part and parcel of a scheme to overthrow the democracy, and that nothing of all this had been done without Alcibiades; the proofs alleged being the general and undemocratic licence of his life and habits.

Later, as the expedition arrived in Sicily, they met an Athenian ship giving orders for Alcibiades to return and stand trial. Thucydides describes an witch-hunt atmosphere back in the city as the democracy sought the culprits.

53There they found the Salaminia come from Athens for Alcibiades, with orders for him to sail home to answer the charges which the state brought against him, and for certain others of the soldiers who with him were accused of sacrilege in the matter of the mysteries and of the Hermae. For the Athenians, after the departure of the expedition, had continued as active as ever in investigating the facts of the mysteries and of the Hermae, and, instead of testing the informers, in their suspicious temper welcomed all indifferently, arresting and imprisoning the best citizens upon the evidence of rascals, and preferring to sift the matter to the bottom sooner than to let an accused person of good character pass unquestioned, owing to the rascality of the informer.

It's important not to overestimate the superstition of ancient peoples. They certainly had many religious beliefs and superstitions, but they weren't naive. When the Hermae were desecrated the first suspicion of most Athenians was probably that it had been done as a political act against the expedition. In this sense though, desecrating the statues was effective on at least two fronts. The first is the one you mentioned, that it was perceived as a bad omen by the superstitious. The second and arguably more important which Thucydides focuses on in the second quote is that it created in-fighting and suspicion within the Athenian democracy. Amongst non superstitious citizens a good modern day conception of how it may have been seen is how people in the present day react when a war memorial is desecrated - even if we are less superstitious than our ancestors, we see these as sacred and any disrespect shown to them is met with calls for the perpetrators to face the full power of the law.