Part 2 of 4
Your Second point seems to be that Burke, prior to his role in the Order of Malta, had been a minor US Bishop who had then removed to a job in Rome where “he did not participate in any adjudications”. There’s a great deal missing in your analysis here.
Cardinal Burke had been (immediately prior to November 2014) the highest ranking US bishop in the Vatican. This changed only when Pope Francis removed Burke from the Prefecture of the Apostolic Signatura at that date. 
Burke also held a string of other important roles. And although he was removed from some of these, he was reappointed in September 2015 to the Curia as a member of the Congregation for the Causes of Saints.
Throughout Francis’s papacy - and whether in or out of specific roles - Burke has always been one of the clear leaders of the Church’s ‘traditonalist’ wing.