Are all problems approached and solved in fundamentally the same way?
What you're missing is that, although all NP-complete problems can be reduced to one another, so it is true that solving one lets you solve them all, that doesn't mean that they're all equally easy, so they're not all "the same problem".

In particular, the reduction takes some amount of time to perform.  Although all problems in NP can be reduced to, say, SAT, that doesn't mean that all problems in NP are as easy as SAT.  Some require very little effort to translate to SAT; others require much more effort. And there's no bound1 on how much effort might be needed, as you look at more and more complex problems within NP.

A way to make this concrete is with the time hierarchy theorem.  That says that, essentially, allowing more time for computations lets you compute more things, and it applies to nondeterministic machines as well as deterministic ones.  So, solving SAT instances takes some amount of time but, if you allow yourself more time than that, you can solve strictly more things.



1 OK, its always a polynomial but there's no bound on the degree of that polynomial.