What would happen if the vote went completely opposite, if the people realized that the police were simply other people who had a bit of specialized training, and the citizenry decided to arm themselves and receive the training necessary so that everyone of them could qualify as a police officer?

If people can't trust their neighbors enough, because the whole bunch of them act criminally, registering guns isn't going to solve any problems. It will only serve to push the bad among them underground, where they will be harder to control than ever.