Securely log in a user without a password using a link?

Yes, you can. However, you’ll need to know either the userid or the username.

Create a link with any of that as a data attribute and make a ajax call with that. On success, you can just reload the page and your user will be logged in.

If you know the user id..

function wpse_304392_login_by_id() {

    $user_id = $_POST['user_id'];

    $user = get_user_by( 'id', $user_id ); 
    if( $user ) {
        wp_set_current_user( $user_id, $user->user_login );
        wp_set_auth_cookie( $user_id );
        do_action( 'wp_login', $user->user_login );
    }

    wp_die(); //Essential for AJAX calls
}

If you know the username..

function wpse_304392_login_by_username() {

    $username = $_POST['username'];
    $user = get_user_by( 'login', username ); 
    $user_id = $user->ID;

    if( $user ) {
        wp_set_current_user( $user_id, $username );
        wp_set_auth_cookie( $user_id );
        do_action( 'wp_login', $username );
    }

    wp_die(); //Essential for AJAX calls
}

Don’t forget do a nonce check to make it secure.