> So I'd like to suggest to write import tests on their module test
> by using _testcapi.run_in_subinterp() for each case.

I like this idea. But I would prefer to factorize the code as much as possible. Like being able to add more checks later without having to modify N files. In test_xxx, I expect something like:
---
class SubinterpreterTests(support.SubinterpreterTestCase):
    module_name = "xxx"
---

Nothing else, all logic would live in a different file, like test.support.

I prefer to use a string for the module. For example, it is possible to import a fresh copy of the module in the current interpreter using a module name.

We can start with a single test which imports the module in run_in_subinterp(), but it becomes possible to add more tests later. For example, import in the current interpreter, import in a subinterpreter, and ensure that the two modules are not the same object.

I see an advantage of putting such tests in each test_xxx file: it prevents to have to import 50+ modules in test_interpreters which would increase its memory footprint, likely have unexpected side effects, etc.

Maybe support.SubinterpreterTestCase should give the ability to override some method. For example, if we need a fresh copy of a module, maybe we should give the ability to provide a list of blocked modules. That's important for "io" vs "_pyio" tests, or "datetime" vs "_datetime", etc. (PEP 399)