Can you confirm you went through these steps in particular?
In any case, are you familiar with your browser’s developer tools? On the network tab and after reloading the authentication page, can you find requests to /api/mfe_context in your LMS instance? What does the server response to that request consist of?
You should be able to find Google listed in a JSON list named "providers". Whether it’s there or not will inform you whether the LMS is instructing the MFE to display the proper SSO button. (In other words, if it’s not there, the problem is in the backend.)