Why are so few natural candidates for NP-intermediate status?
As others have pointed out, it's debatable to what extent the thing you're trying to explain is even true.  One could argue that, in the 60s and 70s, theoretical computer scientists were just more interested in the sorts of problems that turn out to be either in P or else NP-complete.  Today, because of the rise of complexity-theoretic cryptography, quantum computing, lattices, etc.---as well as the simple fact that NP-completeness has become so well-understood---we've become more and more interested in the sorts of problems that turn out to be NP-intermediate.

Still, one could ask: to the extent that the thing is true---that is, to the extent that so many natural search and optimization problems "snap" to being either NP-complete or else in P---to that extent, why is it true?  Here, I think you can get a lot of intuition by looking at an earlier phenomenon from computability: that so many natural models of computation "snap" to being Turing-complete.  In that case, I'd say the explanation is that, once you have a few basic components---a read/write memory, loops, conditionals, etc.---it's hard to avoid being able to simulate a Turing machine, and therefore being Turing-complete.  In much the same way, once your search or optimization problem has a few basic components---most importantly, the ability to construct "gadgets" that mimic logic gates like AND, OR, and NOT---it's hard to avoid being able to encode SAT and therefore being NP-complete.

The way I like to think about it, problems like SAT exert a powerful "gravitational pull" on all the other computational problems in their vicinity, making them want to "snap up" to being NP-complete also.  So, it normally doesn't even require special explanation when yet another problem succumbs to that pull!  What's more striking, and more in need of explanation, is when an (apparently) hard NP problem has some property that lets it resist the gravitational pull of SAT.  We then want to know: what is that property?  Why can't you play the usual NP-completeness trick for this problem, of constructing gadgets that encode Boolean logic gates?  I made up a list of some common answers to that question in this recent CS.SE answer, but (as another commenter already pointed out) there are other possible answers that I missed.