Add rewrite endpoint to author page + pagination

Your code should works. The line:

include (get_template_directory_uri() . '/articles.php');

Needs to have allow_url_include=1 in the server configration because you are trying to include a file via http. Can you check this?

You must know also that template_redirect should be use for a real redirect, a include() may have undesired effects here. I think what you really want is template_include:

add_filter('template_include', 'my_template_include');

function my_template_include( $template ) {
    global $wp->query;
    if ( array_key_exists( 'articles', $wp_query->query_vars ) ) {
        return get_template_directory() . '/articles.php';
    }
   return $template;
}

So here is my code:

function foo_change_author_base() {
    global $wp_rewrite;
    $wp_rewrite->author_base="writer";
}
add_filter( 'init', 'foo_change_author_base', 0 );

function foo_rewrite_rule()
{
    add_rewrite_rule( 'writer/([^/]+)/articles/page/([0-9]{1,})/?$', 'index.php?author_name=$matches[1]&articles=1&paged=$matches[2]', 'top' );
    //If you don't want the last trailing slash change the las /?$ to just $
    add_rewrite_rule( 'writer/([^/]+)/articles/?$', 'index.php?author_name=$matches[1]&articles=1', 'top' );
}
add_action( 'init', 'foo_rewrite_rule' );

function foo_add_vars($vars) {
    $vars[] = 'articles';   
    return $vars;
}
add_filter('query_vars', 'foo_add_vars');

function foo_add_endpoint() {
    add_rewrite_endpoint( 'articles', EP_AUTHORS );
}
//add_action( 'init', 'foo_add_endpoint'); // Commented out
// Edit: add_rewrite_endpoint is not needed when using add_rewrite_rule


// auto-add the articles endpoint to author links
function foo_author_link_filter( $link, $author_id, $author_nicename ) {
    return user_trailingslashit($link) . 'articles/';
}
add_filter( 'author_link', 'foo_author_link_filter', 20, 3);

add_filter('template_include', 'foo_template_include');
function foo_template_include( $template ) {
    global $wp_query;
    //You may need some other checks
    if ( array_key_exists( 'articles', $wp_query->query_vars ) ) {
        return get_template_directory() . '/articles.php';
    }
    return $template;
}

Now go to yoursite.com/writer/author_name/articles/