Just ask people why they vote, and most of them will say things like "It's a civic duty," "Our forefathers died for this, so we shouldn't waste it," "If everyone didn't vote, things would be bad," ...

These views reflect the endorsements of various trusted political figures and groups, the active promotion of voting by those with more individual influence, and the raw observation of outcomes affected by bulk political behavior.

In other words, the common sense or deontological rules of thumb are shaped by the consequences, as the consequences drive moralizing activity. Joshua Greene has some cute discussion of this in his dissertation:

I believe that this pattern is quite general. Our intuitions are not utilitarian, and as a result it is often possible to devise cases in which our intuitions conflict with utilitarianism. But at the same time, our intuitions are somewhat constrained by utilitarianism. This is because we care about utilitarian outcomes, and when a practice is terribly anti-utilitarian, there is, sooner or later, a voice in favor of abolishing it, even if the voice is not explicitly utilitarian. Take the case of drunk driving. Drinking is okay. Driving is okay. Doing both at the same time isn’t such an obviously horrible thing to do, but we’ve learned the hard way that this intuitively innocuous, even fun, activity is tremendously damaging. And now, having moralized the issue with the help of organizations like Mothers Against Drunk Driving—what better moral authority than Mom?—we are prepared to impose very stiff penalties on people who aren’t really “bad people,” people with no general anti-social tendencies. We punish drunk driving and related offenses in a way that appears (or once appeared) disproportionately harsh because we’ve paid the utilitarian costs of not doing so.39 The same might be said of harsh penalties applied to wartime deserters and draft-dodgers. The disposition to avoid situations in which one must kill people and risk being killed is not such an awful disposition to have, morally speaking, and what could be a greater violation of your “rights” than your government’s sending you, an innocent person, off to die against your will?40 Nevertheless we are willing to punish people severely, as severely as we would punish violent criminals, for acting on that reasonable and humane disposition when the utilitarian stakes are sufficiently high.41

The consequentialist justification only happens in obscure corners of geekdom like LessWrong and stat / poli sci journals.

Explicitly yes, but implicitly...?

Just ask people why they vote,

Do you have in mind average people, or, e.g., top 10% Ivy Leaguers ... ?

Just ask people why they vote, and most of them will say things like "It's a civic duty," "Our forefathers died for this, so we shouldn't waste it," "If everyone didn't vote, things would be bad," ...

These reasons aren't obviously deontological (even though they might sound like they are on first hearing). As you say in your comment, timeless decision theory is relevant (transparently so in the last two of the three reasons that you cite).

Even if people did explicitly describe their reasons as deontological, one still wouldn't know whether this was the case, because people's stated reasons are often different from their actual reasons.

One would want to probe here to try to tell whether these things reflect terminal values or instrumental values.

Interestingly, the author also says: "Your decision to vote or not will not affect whether or not other people will vote (unless you are a highly influential person and you announce your voting intention to the world in advance of the election)." This may be mostly true in practice, but not in the limit as everyone approaches identity with you. It seems like this author is a two-boxer based on his statements. He calls timeless considerations "magical thinking."

He may also be a two boxer who thinks that one boxing is magical thinking. However this instance doesn't demonstrate that. Acting as if other agents will conditionally cooperate when they in fact will not is an error. In fact, it will prompt actual timeless decision theorists to defect against you.