I thought the meaning of the exclusion clause was that the government could NOT establish a common religion (congress/legislature etc)?
It is. But the 1st Amendment tacitly acknowledges Congress’ sole authority to establish anything that might be established by explicitly naming things it may not establish.
IOWs what other authority does the government have to establish anything than through a legislative act of Congress? The other two branches, nor any subordinate agency, could establish a state religion, even if it were not prohibited, because they don’t have the Constitutional authority to do so.
Putting up a statue or monument doesn’t constitute such an act in any case. Even if Congress enacted a bill to put a statue of Jesus in the Capitol Rotunda that would not be establishing a state religion. It would be establishing a Congressional decoration of the Capitol.