Login screen config prevents SAML icon to be displayed

Reproduction scenario

1. setup up Nuxeo in a SAML environment like in the documentation here:
   https://doc.nuxeo.com/nxdoc/saml-20-authentication/#integrating-the-nuxeo-platform-with-a-saml-20-idp
2. keep FORM_AUTH in the authentication chain to see the button
3. access Nuxeo login page
4. observe the IdP button is diplayed under the login fields (see withoutBranding.png)
5. configure branding in Studio
6. sync with the server
7. observe the IdP button is not displayed any longer (see withBranding.png)
8. now remove the branding from studio
9. sync the server
10. now try without Studio add the following contribution (it is the same as the Studio one)

      <extension target="org.nuxeo.ecm.platform.ui.web.auth.service.PluggableAuthenticationService" point="loginScreen">
        <loginScreenConfig>
          <bodyBackgroundStyle>url("${org.nuxeo.ecm.contextPath}/img/login_bg.jpg") no-repeat center center fixed #333</bodyBackgroundStyle>
          <disableBackgroundSizeCover>false</disableBackgroundSizeCover>
          <removeNews>false</removeNews>
          <footerStyle></footerStyle>
          <loginBoxBackgroundStyle>none 0 0 null</loginBoxBackgroundStyle>
          <loginBoxWidth>300px</loginBoxWidth>
          <loginButtonBackgroundColor>#0066ff</loginButtonBackgroundColor>
          <logoUrl>${org.nuxeo.ecm.contextPath}/img/nuxeo_logo.png</logoUrl>
          <logoAlt>Nuxeo</logoAlt>
          <logoWidth>113</logoWidth>
          <logoHeight>20</logoHeight>
        </loginScreenConfig>
      </extension>
    

    to the SAML configuration XML file defined server-side.

11. observe when it is inserted before SAML configuration, the login config is not taken into account
12. observe when it is inserted after SAML configuration, IdP button is not visible

Expected behavior: configuring branding should not alter the login screen

Primary analysis

1. the login screen is generated from the page $NUXEO_HOME/nxserver/nuxeo.war/login.jsp
   This is located into those pieces of code:
   https://github.com/nuxeo/nuxeo/blob/master/nuxeo-services/nuxeo-platform-web-common/src/main/resources/web/nuxeo.war/login.jsp#L44

   oginScreenConfig screenConfig = LoginScreenHelper.getConfig();
   List<LoginProviderLink> providers = screenConfig.getProviders();
   boolean useExternalProviders = providers!=null && providers.size()>0;

   https://github.com/nuxeo/nuxeo/blob/master/nuxeo-services/nuxeo-platform-web-common/src/main/resources/web/nuxeo.war/login.jsp#L484

           <% if (useExternalProviders) {%>
           <div class="loginOptions">
             <p><fmt:message bundle="${messages}" key="label.login.loginWithAnotherId" /></p>
             <div class="idList">
               <% for (LoginProviderLink provider : providers) { %>
               <div class="idItem">
                 <a href="<%= provider.getLink(request, request.getContextPath() + request.getParameter("requestedUrl")) %>"
                   style="background-image:url('<%=(context + provider.getIconPath())%>')" title="<%=provider.getDescription()%>"><%=provider.getLabel()%>
                 </a>
               </div>
               <%}%>
             </div>
           </div>
           <%}%>
   

2. after some debugging, the problem is the LoginScreenHelper.getProviders() method returns an empty list of providers when the login screen config is contributed
   https://github.com/nuxeo/nuxeo/blob/master/nuxeo-services/nuxeo-platform-web-common/src/main/java/org/nuxeo/ecm/platform/ui/web/auth/service/LoginScreenConfig.java#L63
3. this should not be the case because the SAMLAuthenticationProvider is actually registering itself here:
   https://github.com/nuxeo/nuxeo/blob/master/nuxeo-services/login/nuxeo-platform-login-saml2/src/main/java/org/nuxeo/ecm/platform/auth/saml/SAMLAuthenticationProvider.java#L245

   
               LoginScreenHelper.registerLoginProvider(parameters.get("name"), parameters.get("icon"), null,
                       parameters.get("label"), parameters.get("description"), this);

4. it seems the LoginScreenConfig is reset for some reason but I cannot pinpoint the location