However, I don't see where "value of information" shows up in this framework anywhere. Do I need to distinguish some choice nodes as "information gathering", and apply a default strategy to them as "don't gather any information", and then compute value of information as the difference between best I can do with my eyes closed and the best I can do flat out?

Think of VoI as going in the reverse direction. That is, beforehand you would have modeled your test outcome as a nature node because you didn't consider the option of not running the test. Now you stick in a choice node of "run the test" that leads to the nature node of the test output on the one branch, and the tree where you don't know the test output on the other branch. Like you suggest, you then use the work-backwards algorithm to figure out the optimal decision at the "run the test" node, and the difference between the branch node values is the absolute value of the VoI minus the test cost.

What if there is no natural partition of some actions as information gathering and not-information-gathering?

Then VoI won't help you very much. VoI is a concept that helps in specifying decision problems- building the tree- not computing a tree to find an optimal policy. It suggests modeling information-gathering activities as choice nodes leading to nature nodes, rather than just nature nodes. If you've got a complete decision problem already, then you don't need VoI.

I should point out that most tests aren't modeled as just information-gathering. If a test is costless, then why not run it, even if you throw the results away? Typically the test will have some cost, in either utility or prospects, and so in some sense there's rarely actions that are purely information gathering.