I am not sure that the Internet has changed the way we think about research, or anything else for that matter, but I would agree that the problem of thinking "within a context of no context" raised by George W.S. Trow is real and worth thinking about some more (see And Now He's Dead). But it's a problem that is around at least since Herodotus. If there is a context of (all) contexts, we have not clearly identified it yet. On the other hand, contextualization alway presupposes some kind of larger context, even if we usually do so unreflectively.
I liked the quote by J.D. Marshall about the sources of research in The Tyranny of the Discrete:
If the student remains absorbed in original sources, he or she will be content to reproduce information from these sources, which he or she will regard as having special historical validity. In other words the information itself will become a substitute for history; the discrete fact itself becomes pseudo-history.True, but "coherent," immediate," "imaginative" and "telling" contexts presuppose wider contexts, and that is what we seem to have "lost." Insofar as you can only "lose" what you once had, the question should be whether , or perhaps better, in what sense we "had" such contexts.
Students ... are more likely to develop the notion that crucial or important historical information comes from manuscript sources only. This leaves them unable to appreciate an argument which uses a variety of different sources to provide evidence. They are also prone to develop a prejudice to the effect that facts are more important than arguments, and the verification of facts represents almost the sum total of local historical activity. ... Local history, like any other kind of history is meaningless without coherent, immediate, imaginative and above all telling context
I would argue that we had them only insofar as we made them, which is not to say necessarily that they were arbitrary.
The Internet may not encourage thinking about larger contexts, but what else is new?