It is extremely easy to convince yourself that the current system works. Numerous people volunteer to work in election monitoring every year, and any person who is not sure can take a day or two off work to do so at their next election.

Plus, the system overall is dead simple, first grade math skills are enough to understand it: we just count the votes in every precinct, and sum up the votes later up. No hashes, no smart group theory schemes, nothing complex.