Automatically set Homepage

You want to use update_option for that.
https://developer.wordpress.org/reference/functions/update_option/

function set_home_page($post_id) {
    update_option( 'page_on_front', $post_id );
    update_option( 'show_on_front', 'page' );
 }
add_action( 'publish_post', 'set_home_page' );