Logout redirect via page template without confirmation?

Question

I'm trying to redirect users when they visit my page template template-logout.php, but I keep getting the Wordpress "Do you really want to logout?" question despite wpnonce being generated. What am I doing wrong?

Here's an example of the type of URL being generated by my function below when I reach that WP confirmation page:

https://localssl:8890/wp-login.php?action=logout&redirect_to=https%3A%2F%2Flocalssl%3A8890&_wpnonce=752dc07192

CODE

template-logout.php

<?php 
/**
 * Template name: Log Out
 */

get_header(); 

// Logic handled via functions.php

get_footer(); ?>

functions.php

// Redirects for template pages
add_filter('template_redirect', 'osu_redirect_on_logout');
function osu_redirect_on_logout() {
    if ( is_page_template( 'templates/template-logout.php' ) ) {
        $logout_url = wp_logout_url( home_url() );
        wp_safe_redirect( $logout_url ); // Logout and Redirect to homepage
        exit;
    }
}

** EDIT **

Answer below by Gabriel is the better way of doing this, I was getting muddled with the redirection side of things. To make sure this method redirects to the homepage, this was my final code:

// Redirect to homepage when using wp_logout();
add_action( 'wp_logout', 'osu_redirect_homepage_after_logout');
function osu_redirect_homepage_after_logout(){
    wp_redirect( home_url() );
    exit();
}

// Redirects for template pages
add_filter('template_redirect', 'osu_redirect_on_logout');
function osu_redirect_on_logout() {
    if ( is_page_template( 'templates/template-logout.php' ) ) {
        wp_logout();
    }
}

Answer 1

If you are just trying to logout a user and redirect them back to the homepage, could you just use the wp_logout() function?

So instead of this:

if ( is_page_template( 'templates/template-logout.php' ) ) {
    $logout_url = wp_logout_url( home_url() );
    wp_safe_redirect( $logout_url ); // Logout and Redirect to homepage
    exit;
}

Your logic could just be:

if ( is_page_template( 'templates/template-logout.php' ) ) {
    wp_logout();
}