Yes, I cannot nor would I argue the fact that the founding of this country was primarily by Christians or at least those who claimed to be Christian. Christians who in many cases wanted the government out of the church business, though not all agreed and many states had for a period of time state supported churches.
The statement "I believe the term "separation of church and state" puts no limits on religion in the state, but merely refers to the state's responsibility to refrain from exerting authority over ecclesial bodies." I believe falls short of the full meaning of the First Amendment.
Any relationship cannot be one-sided such as you suggest. To say the government can't exercise authority over what our church teaches on the one hand then to expect the government to bend to the desires of a church(es) is disingenuous at best and still leaves unanswered, which church? If you say Christian, which denomination? If you say Baptist, which version? Then you have to come up with an explanation as to why a Baptist American is more important then a Jewish American (or any other nonBaptist American) who though not as populous at the founding of this country, were still very influential in the entire process. By maintaining a neutral stance on religion by not endorsing, helping or interfering freedom for all is guaranteed
A mechanism has been put in place for the religious people to "influence" the government, the voting process. The ballot box is where all Americans stand on level ground. One person one vote.