Of course it's hyperbole. However, the fact remains that a legislator's religious belief is a major factor determining why people vote for him/her. People want to see legislators who mirror their own religious values in the government. Once in government, legislators are in a position to impose religious dogma through legislative means... and many have done so. When this occurs, the beliefs of a majority religion intrinsically colors the legislative process. When every session of congress opens with a Christian prayer, then I believe that I am correct in my presumption that said religion is being recognized by the governmental process in a way that I disapprove of. My government is asking a specific religion's diety to "guide" them. When I sit in a church and the preacher tells me that if I do not vote for candidate A over candidate B then I will go to hell, the church is most assuredly pressing itself into the governmental process. When candidates who are non-religious have no chance of being elected to state or national public office, then the church is again is shown to be the crux of who is and is not allowed governmental power.
Therefore, the statement that the church IS the government, hyperbole though it may be, is not necessarily incorrect.
As for who is an "authentic Christian" and who is not, that's not for me to say. That would be between them and whatever God they believe in. I strongly... strongly... support the freedom of all Americans to worship as they please. However, given the clashes between religious groups lately, including the Christian-led anti-mosque silliness of the past years and the histrionic reaction of so many to anyone who does not believe at all, I'm thinking a hell of a lot of Americans are pretty selective about who is and is not allowed such freedoms.
This is one reason I stay away from religious forums. I see no need to intrude upon people's discussion of their own beliefs. Also, I note the reaction to those who do not share them. When a viewpoint you could not agree with mixed religion and government you, according to your own words, had difficulty even being polite to me although we have gotten along quite well in the past and agreed with each other frequently! People do not like their religions to be challenged. I do not challenge them. I simply do not like them being a cornerstone of the government, and those who run it.