I had the same question and the same issue.
But with some changes to your code, you can hide the WordPress login and password fields, and still keep the social logon button.
Just add the code below to your theme’s functions.php
file
/**
* This hides the login and password fields so only the social login button is visible
*/
add_action( 'login_head', 'wpse_121687_hide_login' );
function wpse_121687_hide_login() {
$style="";
$style .= '<style type="text/css">';
$style .= '.login label{ display: none }';
$style .= '.login .button.wp-hide-pw .dashicons{ display: none }';
$style .= '.login form .input{ display: none }';
$style .= '.login form .forgetmenot{ display: none }';
$style .= '.login #nav a, .login #backtoblog a { display: none }';
$style .= '</style>';
echo $style;
}
There is probably some more tweaking that could be done, like removing the large LOGIN button, but I’m still working on it.
Any advice on improvement from an expert would be appreciated!