first off there is a huge difference between electing a leader and having one forced on you.
Second if you actually look at what young rulers have done it's not very much. They are more of a figure head until they come of age. They are groomed if you would and they have strong advisers who instruct them how to rule until they are usally in their 20s. Then they are given more power.
Perfect example is Cleopatra and her little brother Ptolemy XIV of Egypt. Cleopatra actually ruled while her brother Ptolemy was just a figure head although he did have the power to send many of his servents to their death and he did so, he had no actual say in the running of the country or it's people.
Even Alexander the Great wasn't proclaimed King until he was 20 and even then he didn't have full power his mother did a lot of the decision making.
Yet they were still thrust upon the throne and given power(no matter how little) were they not? Thats where it really divides though and where real life logic has no purpose. For all we know in the SWU they run things differently and each planet has its own set of laws, so I don't see how its much of a stretch Padme could hold power.