How to display only sticky posts on my automatically generated front page?

WordPress saves sticky posts inside the option named sticky_posts. You can retrieve them via get_option('sticky_posts').

With that knowledge it is possible to change the query to only query for sticky posts, you usually do so via the pre_get_posts hook.

add_action('pre_get_posts', 'WPSE_home_only_stickies');
function WPSE_home_only_stickies($query) {
    // only run for homepage & main query
    if ($query->is_home() && $query->is_main_query()) {
        $sticky_posts = get_option('sticky_posts');
        $query->set('post__in', $sticky_posts);
    }
}