Why was this question put on hold? It seems to be fairly specific and it is about mathematics. You could argue that it is slightly meta-mathematical; would it then be appropriate on meta MO?
I do not think there is much evidence to support your assertion that your question is specific. You ask whether there is research on "in which mathematical universes does Grothendieck's approach succeed", and we are left with the rather formidable tasks of unpacking the terms "mathematical universes", "Grothendieck's approach", and "succeed".
When people asked for clarification in the comments, we see that the definition of "mathematical universes" is unclear, but it seems you are partial to considering arithmetic systems too weak to prove Godel's incompleteness theorem. Perhaps you would have better luck considering nonstandard models that can actually prove incompleteness. Your definition of "succeed" seems to be "thousands of pages of monographs", which is a bit stringent for work on hypothetical worlds. Finally, "Grothendieck's approach" has been reduced to something like being careful when defining things, and this hardly seems to be unique to Grothendieck.
One point I'd like to emphasize is that when you ask a question where an answer might require non-trivial effort, you should take care to remove as much ambiguity as you reasonably can, and you should at least give the impression that you will take serious answers seriously. If you haven't done your homework, you shouldn't expect others to invest time and energy helping.