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 ...
I believe that I can give a partial answer to your question. Looking at some of the profiles that you have listed, they seem to be all unregistered users. I used to do something similar (but with only 4-5 names) in 2012, at my beginnings on MO: not in bad faith, but because I was viewing my asking questions on MO as only a temporary thing. Plus, I was ...
I think it is reasonable to be concerned, if only for the fact that we do not know what the ramifications of this behaviour are.
However, I think it is better to describe the behaviour and call out what might be problematic about the behaviour. Right now, it seems we are getting OK to good quality material for the forum, and we have the ability to improve ...
Ilya Bogdanov has answered my question Graphs with only disjoint perfect matchings
on certain coloring in graphs, that emerged through research in quantum physics. This answer has inspired quite a bit of research:
Quantum Experiments and Graphs (arXiv - cited Bogdanov's MO answer)
Quantum experiments and graphs II (arXiv)
Quantum experiments and graphs. ...
Please take note of parallel efforts, usually more narrowly focussed:
(1) The Open Problem Garden,
originally concentrating on graph theory, but since expanded.
(2) The Open Problems Project,
focussed on discrete and computational geometry.
This area is under active development through the
Workshop on Open Problems and ...
This MO question "Property $\Gamma$ in terms of correspondences" led us to answer two old open problems and to push further a third more recent result:
Jon Bannon, Amine Marrakchi, Narutaka Ozawa. Full factors and co-amenable inclusions, arxiv/1903.05395.