gh-106320: Remove _PyDict_GetItemStringWithError() function (#108313)

Remove private _PyDict_GetItemStringWithError() function of the
public C API: the new PyDict_GetItemStringRef() can be used instead.

* Move private _PyDict_GetItemStringWithError() to the internal C API.
* _testcapi get_code_extra_index() uses PyDict_GetItemStringRef().
  Avoid using private functions in _testcapi which tests the public C
  API.