What is it but xxx
Is a more poetic way of saying
It is no more than xxx
The author is saying that "American government" is considered by some to be a carefully crafted construction but (says the author) it is only a pattern of behaviour that we repeat from generation to generation; it is nothing more than a tradition.
The use of though and but is valid because they both introduce separate qualifications to the main idea.
The American government is a tradition endeavoring to transmit itself to posterity.
First the author qualifies tradition. Unlike some traditions that are many hundreds of years old, American (US) government dates only to 1776.
The American government is a tradition, though a recent one, endeavoring to transmit itself to posterity.
Second the author says that as the tradition transmits from generation to generation changes occur. So the transmission from one generation to the next loses integrity. So the but is explaining that the endeavour to transmit is not completely successful.
The American government is a tradition, though a recent one, endeavoring to transmit itself to posterity, but losing some integrity.
The each refers to the instant; the passing of time leads to loss of integrity.