It is a big deal if they vote. Not a big deal for the students personally if they leave the US permanently, but it is a big deal for citizens who have to live with the fact that those noncitizens may have altered the election.
The consulate doesn't have to say anything about AOS or citizenship. Voting or registering to vote for an election where citizenship is required can result in losing their F-1 status, if the incident is discovered while they still have F-1 status.
Well, the bottomline is that government think it is citizens' and noncitizens's responsibility to know how law works. I don't think this issue is only related to immigrants. Even many good citizens can end up in big trouble simply because they do not know what they do is illegal.