I think what you are asking is whether Stanford will use the test without essay at all to determine admission, including, particularly, when it provides the better section scores. Here is Stanford’s discussion of what you need to send, how, and what will be used: https://admission.stanford.edu/apply/freshman/testing.html. It specifically answers practicably every question one could ask except the one you are asking. You self-report scores when applying and self-report scores of all tests. Stanford says a test without essay “will not complete the testing requirement” but it does not say what happens if you submit both an SAT with essay (and thus “complete the testing requirement”) and an SAT without essay, except it mentions that it superscores SAT tests and, in your application, you are to first list your highest section scores.
Bottom line: the only way to get a definitive answer to your inquiry is to ask Stanford, but one interpretation is that it might use the test without writing. The way it has you do self-reporting – listing highest section scores followed by next highest section scores from multiple tests – indicates that you should report the section scores of the test without writing, and by listing scores as Stanford requests, withut specifying which section scores come from which test, Stanford will not even know which math and evidenced based reading/writing scores come from which test.