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I've figured out what the primary problem is on these platforms: os.stat() returns st_mtime and st_atime values with nanosecond resolution, but without a corresponding utimensat(), we can only affect time with microsecond precision via utimes().
Therefore, the following logic is faulty:
 def _test_utime(self, filename, attr, utime, delta):
 # Issue #13327 removed the requirement to pass None as the
 # second argument. Check that the previous methods of passing
 # a time tuple or None work in addition to no argument.
 st0 = os.stat(filename)
 # Doesn't set anything new, but sets the time tuple way
 utime(filename, (attr(st0, "st_atime"), attr(st0, "st_mtime")))
 # Setting the time to the time you just read, then reading again,
 # should always return exactly the same times.
 st1 = os.stat(filename)
 self.assertEqual(attr(st0, "st_mtime"), attr(st1, "st_mtime"))
 self.assertEqual(attr(st0, "st_atime"), attr(st1, "st_atime"))