Work in an area or work on problems? I have a question about two possible career paths, for which I was unable to come up a better title. Let me explain what I mean:

Path 1, Working in an area: By this I mean, making a career by adding to the knowledge of a field of study. This may include sorting out open questions in that field or identifying new issues or pushing the boundaries of existing knowledge. This typically involves having a larger perspective and understanding of the field and its relevance to the world. 

Path 2, Working on problems: By this I mean making a career by solving a series of specific challenging problems not necessarily belonging to a common field of study. Here one only attempts to understand enough about the problem at hand to solve the problem, but does not show an interest in developing the area as such.

Working in an area requires one to have a broader vision, scholarship and commitment to the development of the area. Working on problems does not involve commitment, but requires one to repeated invest oneself in learning about a new area. By working in an area one can encounter a degree of monotony. By working on problems, one can potentially find new challenges at every juncture.

So my question is: career wise, what is a better option? Specifically, which of these kind of academics are more valued by the community? What, if any, are pitfalls of these paths?
Meta question: is path 2 a path at all or do all academics eventually settle into path 1 after spending some time on path 2?

Edit: I guess the key difference between the two paths is that path 1 leads one to become an "expert" with extensive knowledge in a particular area. Path 2 exposes one to a variety of problem situations and makes one a better problem solver, though it may not make one an expert in any field. For the purpose of this question, you may take the area to be a well studied field such as, say integer programming, which has some long-standing open problems, but is not necessarily so young that it allows for a variety of research opportunities.
Without knowing anything about the specific area, I'd say that you've set up a false dichotomy. I don't know if anyone ever consciously does one or the other exclusively. Sometimes you work bottom up (i.e path 2 -> path 1), and sometimes you work top down (path 1 -> path 2). Both these "paths" should be dimensions of your research.