5. The problem isn't chicanery in the counting of the votes.
The election is conducted by the National Labor Relations Board, not by the employer, so substituting the state government wouldn't accomplish much.
In any event, the count is generally thought to be accurate. The problem is in the run-up to the vote, where there are two major categories of issues. First, the law gives management significant advantages in the campaigning, such as the ability to require workers to attend anti-union meetings. Second, if management isn't satisfied with its legal advantages, it can get away with using illegal methods, such as firing pro-union employees.
One advantage of card check is that the union can avoid some of these problems with a stealth campaign. Organizers can privately talk to workers and get them to sign representation cards. In the ideal case, management has no idea that there's an organizing drive afoot until the organizers proudly present them with the signed cards. Under current law, without card check, management can demand an election and then swing into action to poison the waters before the vote.