What does the fact that aliens generally are not and have never been allowed to vote show about American beliefs?
There is no single correct answer to this. It is possible to draw at least two conclusions about American beliefs, depending on one’s own attitudes.
On the one hand, we can see this restriction as evidence that we Americans are hypocritical. We say that people should have a say in the way that they are governed. People who are in the US, but who are not citizens, are subject to our laws (unless they have diplomatic immunity) just like everyone else. Therefore, it would stand to reason that they should be able to have a say in those laws. The fact that we do not allow them to vote, in this view, shows that we are hypocritical and xenophobic.
On the other hand, we can see this restriction as evidence that we are truly committed to the ideas of personal choice and self-determination. In our history, people who enter our country lawfully and then follow the correct procedures can choose to become Americans. Of course, there are many who would like to be Americans but cannot because of limits on immigration, but becoming American is in essence a choice. From this point of view, our restrictions on who can vote show that we truly believe in giving people choices. If you choose to become an American, you can vote. If you choose not to become an American, you are choosing not to vote because you are choosing not to truly become one of us.
All nations have laws regulating who may vote and who may not vote in elections. First and foremost among these laws is the requirement for citizenship. What America's voting prohibition says about America is that, like other nations, America wants internal issues solved and voted on by citizens. Though some nations have more particular voting rights laws, like Hungary and Indonesia:
Hungary18 years of age, 16 if married; universal (CIA World Factbook)
Indonesia 17 years of age; universal and married persons regardless of age (CIA World Factbook)