There is a lot of hate for the big 3, not just for cricketing reasons.  But if we are honest , they are well clear of the rest of the other cricketing nations in terms of quality and consistency. 
Whenever 2 of the 3 play each other in any format, I always find the cricket to be quality and extremely competitive.  I enjoy watching them do battle as well. The players seem to want to go that extra mile when playing each other.
The honest truth is they produce the best games and quality of cricket when 2 of the 3 face each other.
		
		
	 
Definitely yes.
This "Quality" which perhaps cannot be captured in statistics and numbers, but is evident to mostly veteran fans. 
The tape ball generation who mostly believes that the only stroke in cricket is a 6, may not have much of an idea of what we are talking about here.
The class of batting is specially different in big 3.
The way they play proper cricket strokes in the cover area, inside out strokes, drives, and running between the wickets, is different.
Albeit, the recent introduction of Babar Azam, there is hardly anyone outside big three who displays much of that high class. Cow corner slog is the trade mark shot of many non-big 3 batters.
Also,
many big 3 specialist bowlers (specially England and Australia) have proved it at various times that they can step up and play vital BATTING innings when the top order fails or needs help; however, in rest of the teams, the pace bowlers bat down the order as if they are playing cricket reserved  for mentally challenged. Many don't even seem to know how to hold the bat.?
The big 3 has two other edges over the rest.
First, the fielding standard. It's definitely at a different level. The clean pick and throw inside the circle, and diving/catching on the boundary (NZ makes it here as well), is at a different level.
Second, the professionalism. Specially in England and Australia. When a batsman gets run out, he gracefully walks away. 
In other teams, at times there is a little animated verbal scuffle where one tries to blame other before walking away. It looks ******** but that's how run out are usually dealt with. Their professionalism throws in the white towel when the sentimentalism steps in.