Also worth noting that when I click on the Log in with SAML button on the login page, it attempts to login with a Loading page and then just comes back to the Log in with SAML button. We have double/triple checked all the settings in the IIS reverse proxy and Azure AD/SAML configuration. Are there other configurations that are needed to get this to work?
That suggests to me a misconfiguration. Per the docs:
verify that the configuration is correct by clicking on Test Configuration . A SAML login will be initiated and useful information about the SAML response obtained from the Identity provider will be returned.