How do I force “users must be registered and logged in” on subsites?

Here is what you do to force registration on sub sites:

/**
 * Require registration for comments on subsites too.
 */
function ourcomments_pre_option_comment_registration( $value )
{
        return 1;
}
add_filter( 'pre_option_comment_registration', 'ourcomments_pre_option_comment_registration', 10, 1 );

To only show social buttons for login something like this can be used:

/**
 * Hide login fields. Only social logins.
 */
function ourcomments_only_social_logins_login_form() {
?>
  <script type="text/javascript">
(function() {
    // Some things seems to be added later, hide them. And display site name.
    var cssHide = "";
    cssHide += "div#login h1 a { background:none; text-indent: 0; font-size: 30px; font-weight: bold; width: auto; height: auto; color: black; }";
    function checkAndRemove(el) {
        if (el) {
            var elP = el.parentElement;
            if (elP && "P" !== elP.nodeName) elP = elP.parentElement;
            if (elP && "P" === elP.nodeName) {
                elP.parentElement.removeChild(elP);
            }
        }
    }
    checkAndRemove(document.getElementById("user_login"));
    checkAndRemove(document.getElementById("user_pass"));
    // checkAndRemove(document.getElementById("rememberme"));
    cssHide += " div#login form#loginform p.forgetmenot { display:none }";
    // checkAndRemove(document.getElementById("wp-submit"));
    cssHide += " div#login form#loginform p.submit { display:none }";

    // Remove lost password (looks easily breakable...):
    var navLostPassword = document.querySelector("div#login p#nav")
    if (navLostPassword)
        navLostPassword.parentElement.removeChild(navLostPassword);
    else {
        console.log("p#nav not found");
        cssHide += " div#login p#nav { display:none }";
    }

    // For Nextend:
    var nextendOr = document.querySelector("div#login form#login-form h3");
    if (nextendOr)
        nextendOr.replaceChild(document.createTextNode("Please login!"), nextendOr.firstChild);
    else {
        console.log("h3 not found");
        cssHide += " h3 { display: none; }";
    }
    var s = document.createElement("style");
    s.appendChild(document.createTextNode(cssHide));
    document.body.appendChild(s);
})();
  </script>
<?php
}
add_action('login_form', 'ourcomments_only_social_logins_login_form', 1000);
add_action('register_form', 'ourcomments_only_social_logins_login_form', 1000);
add_action('bp_sidebar_login_form', 'ourcomments_only_social_logins_login_form', 1000);

Since the plugins for social buttons do not place those on sub sites you probably want to login through the main site:

/**
 * Login through main site.
 * Mostly a copy of the wp_login_url in general-template.php.
 * Fix-me: move to plugin.
 */
function ourcomments_network_login_url($login_url, $redirect) {
    $net_login_url = network_site_url('wp-login.php', 'login');

    $my_redirect = $redirect;
if ( empty($my_redirect) )
            $my_redirect = $_SERVER['REQUEST_URI'];
if ( !empty($my_redirect) )
    $net_login_url = add_query_arg('redirect_to', urlencode($my_redirect), $net_login_url);

    // We do not have the $force_reauth arg but we know how it looks in the url.
    $values = parse_url($login_url);
    $query = $values['query'];
    parse_str($query, $output);
    if ( $output['reauth'] )
    $net_login_url = add_query_arg('reauth', '1', $net_login_url);

return $net_login_url;
}
add_filter('login_url', 'ourcomments_network_login_url', -100, 2);

This can be added to a themes functions.php. Or put in a plugin (which I did not have time to do yet).

Maybe I should add that a malicious user could still try to get in another way. This is just to make normal users more comfortable.