The K of C on the local level doesn't know what their leaders are doing.  It's just like most Catholics don't know what their bishops are doing.  Bishops are supposed to be shepherds.  But many live like the nobility of long ago.

The K of C's leadership operates just like any other 'secret' society.  In fact, that's why they and other Catholic secret societies [like Opus Dei] were begun.  
In the late 19th & early 20th Centuries, Catholic men who were professional, were invited to join the Masons. So the K of C was the Catholic alternative. So men just like their 'secret societies'.  The higher up in position a man goes, the more little secrets he knows.  

I suggest that you lay too much weight on influence and money. In Matthew's gospel---Jesus throwing out the money-lenders, disrupted the Jewish priests' status quo and their collaboration with the Romans.  Many early Christians also fled to the Desert during the Persecutions---to live an ascetical life---sans influence.