some (not all, or even a large %) of those claiming to be non-citizens are likely using that as a fast way out of jury duty. So more verification would be needed.
Yes, you are correct, but there’s plenty to be caught.
http://www.freerepublic.com/focus/f-news/3481550/posts
NBC2 Investigates: Voter Fraud (Florida: illegals voting, again)
Two elections supervisors are taking action after an NBC2 investigation uncovers flawed record keeping and human error allowing people who are not citizens of the United States to vote.
Voter fraud was so bad in 2012 that the Miami Police Benevolent Society put up dozens of billboards and radio-ads offering a $10,000 reward for information leading to convictions for vote fraud. They have nailed a few big-time operators.