Years ago I had something explained to me and it has formed a basis for how I read a passage in scripture.  It goes like this:  We have the United States of America and most people when they travel to the United States, usually state that they are visiting the country, not their final destination point.  But if they were going to be visiting so to be able to see the city where political decisions were made and enacted, they would say they were going to Washington D.C.
 
To follow this analogy, one could say that a saved person has the title of belonging to the Kingdom of God. This could relate to earthly as well as heavenly, but for this discussion, I believe scripture is talking about the spiritual aspect that we can only be a part of through the rebirthing of our spirit.  Now we have access to this kingdom or realm.  But if you want to get to the governing place within this realm; where the throne is, where decrees are made and carried out, then you would want access to the city...or the Kingdom of Heaven.  The ruling City of Heaven.  The fact that Matt. uses the term exclusively would be because Matt. is presenting Christ as King.  The other gospels present him from a different perspective where his Kingly role isn't the main focus.
 
Maybe this is just another facet to both of your discussions.