When I vote, I get the moral right to complain about other peoples' votes, and therefore to complain about the actions of the government these votes elect.

I see complaining as a basic human right. Saying that people who don't vote shouldn't complain is an effort to eliminate a major source of feedback about how a society is going.

Saying that people who don't vote shouldn't complain (or possibly shouldn't be listened to by voters) seems to me like a claim that was trumped up to get people to vote. Voting makes relatively little difference. How about "people who don't vote in primaries shouldn't complain"? People who don't research their votes shouldn't complain? People who don't take an active part in politics by researching and then trying to influence other people shouldn't complain?