You are right on both accounts.
If we use the government structure test to rule out Latin America as Christian Western societies, then we must a fortiori reject any notion whatever of the United States as a Christian society, for the structure of American government is much more overtly and militantly anti-religion than most of Latin America
True enough, if we exclude Venezuela, Cuba, and Brazil from the mix. However, the formal structure of American government, as expressed in the original intent of the Constitution, was far from irreligious. Until the 20th Century, the general consensus was that the United States government was established on Biblical principles. Further, Anglo-American common law was to a great extent based on Biblical concepts of justice. Latin American law, like that of continental Europe, was based on the Napoleonic Code, which largely drew its inspiration from classical Greco-Roman precedents.
American government in 2005 differs greatly from the ideals of the Founding Fathers and the practices of the first 125 years of American independence.
You're missing a step in the development of the Civil Law, incorporated uniformly across Europe by the French Civil Code, the famous "Code Napoleon".
And the step you are missing is that Civil Law was not "rediscovered" by Napoleon, and didn't come down to the modern European world DIRECTLY from the Romans and Byzantines. It came, rather, through an intermediary: the Canon Law of the Catholic Church. Civil Law was taught at Bologna and Paris, in the Catholic Universities, as part of the unicity between secular and ecclesiastical law. And indeed the specific inspiration for the Civil Law, THE origin of most of it, was the Corpus Iuris Civilis of Justinian. Justinian's Code has two features about it which are crucial. The one is that it codifies most of what is recodified in Canon Law and Civil Law. Napoleon's contribution was not to create a new legal regime for Europe, but to unify the different branches of a very old civil law. The second is that Justinian's Code itself was a thoroughly Christian law code. Justinian's whole legal enterprise in his Code was to take the old, pagan Roman civil law and improve its harshness by Christian principles. Thus, principles of equity were introduced which were quite foreign to pagan Roman mechanical legalism.
So, drill down through the Napoleonic Code, and you will find the Canon Law, which was expressly Catholic and which was the practical law of equity governing all of Europe (England included - the Common Law extended not to equity and the ecclesial courts or Chancery, which handled some of the most crucial legal affairs of people: marital law was not Common Law but ecclesial, which is to say Canon, in England as everywhere else in Europe, at least prior to the Reformation). Drill down through the Canon Law, and you will find Justinian's Code, which was an expressly Christian undertaking (in a way that Napoleon's Code was not).
I agree that the original intent of American government was not irreligious. But then, it would have been surprising indeed, revolutionary, for there to have been ANY irreligious, non-Christian-based government in the European world in 1776. The French "Declaration of Rights of Man and of Citizen" reposes on God-given rights as well.
The violent passions of the Terror swept away the original Providential belief of the mild revolutionaries who first led France into change. When they stepped down, really quite unfortunately, the field was open to radicals, who then resorted to violence against every edifice that could challenge their power, including the nobility and the Church and, really, anyone else who was organized and not of their party.
In looking at the relative state of national laws, I maintain that the most important things are the best indicators of the state of the national soul of a country. Those Christian countries that continue to outlaw abortion are the ones that adhere closest to the truth on the things that really matter...at least in my opinion.