As to secret ballots, it is a nice bit of privacy and it is a protection from a government that is moving towards tyranny. Look at how our government has been using the IRS to target conservatives and libertarians and NSA to spy on journalists.
Usually there is a period of time where a government moves from not being a tyranny towards being a tyranny - usually a period of years or even decades. Once a government becomes a full-fledged tyranny then voting is not an effective expression of people's choices - it is either ended altogether, or it becomes a total fraud - like the former USSR's voting which was required, but never offered the choice of a non-communist opponent.
It is the inbetween period of time where an open ballot would be a danger. It would give the government a way to target those who oppose them. I don't think it is a very long-lasting period because the same government that would abuse an open ballot would also engage in voting fraud. Before a government reaches that state, it simply wouldn't matter. After that stage is passed it is too late to look at the particulars of voting practices as a way to save the country.
The issue of tyranny is probably something we should look at more carefully. When should we declare that we are more oppressed than free? When is it reasonable to start using force to 'water that tree of liberty' Jefferson spoke of? On one level the question always appears as an individual choice. And then it is a choice driven by the individual's context which can vary considerably from person to person. But what can we look at as general guidelines?
Rand spoke about freedom of speech (and assembly) and freedom of the press - I don't remember, but she may have included the absence of significant voter fraud. If so, I can certainly see that approach. When the citizenry can no longer speak or write about the changes they believe are needed, or to assemble to urge their positions and generate support, and they can not vote in a way that lets them change the politicians, then choice is no longer politically viable and only force is left.
There has to be a connection between the choice of the individuals and the way their country is governed if we want to call it representative, and say that government's purpose is to protect choice.
I agree (to an extent) with what Mr. Howard said about very, very few rules. Individual rights is the moral base for laws. The constitution should be a much tighter and more effective limit on the laws that government can make. There need to be many detailed laws, but only in the sense of properly describing variations of those few things that people can't do. For example, there are many ways to initiate force or to threaten to initiate force, and those variations need to be described (extortion, mugging, armed robbery, assault, man-slaughter, murder, etc.) Otherwise you can't differentiate actions in an understandable, objective fashion so as to know what to prosecute and what isn't a violation of rights.
And things like contract law need to be spelled out.
But if it isn't directly arising from individual rights, then there should be no law. And the value of stability (the same set of rules persisting unchanged for generations) is so true.
We have become a country where laws are changed annually or monthly or even daily as part of a partisan squabbling, as means of implimenting ideologies, as means of gaining competitive business advange achieved via lobbiests, as the drive of agency regulators to control nearly every aspect of everyone's life, or just as political kabuki dances put on to sway votes in the next election.
Much of society acts as if only we had enough laws we would achieve utopia - "There should be a law..." is the common refrain. The peculiarity is that the more laws we have, the less we become a nation of laws, and more a nation ruled by whims of elites.