Putting content into header.php without using wp_head

Removing wp_head will have some negative effects as it won’t allow you to use virtually 90% of the plugins on wordpress.org. Also some of that code you are seeing only shows up when you are logged in (unless your theme is specifically adding it). For example, the dashicons, open sans, and admin-bar stylesheets and inline styles at the bottom are all there so you can use the admin bar on the frontend.

Here are some options.

1. Custom header – This is a safe alternative if you don’t want wp_head in some of your pages, but still want to be able to use it and other wordpress plugins on other templates.

1. Create a file header-custom.php in your theme folder that doesn’t include wp_head
2. In your custom page template call that header using <?php get_header( 'custom' ); ?>
3. If you wan’t to be able to hook into this still with through your plugin you can use a custom action instead of wp_head. Basically all you have to do is add <?php do_action('custom_head'); ?> and then in your plugin hook into that the same way you would wp_head.

Example

function my_plugin_custom_head_action() {
    // do stuff here
}
add_action('custom_head', 'my_plugin_custom_head_action');

There’s also a plugin that makes this process of using custom hooks/actions very easy.

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2. Remove stuff you don’t want from wp_head

// Remove emojis
function disable_emojis_wp_head() {
    remove_action( 'wp_head', 'print_emoji_detection_script', 7 );
    remove_action( 'admin_print_scripts', 'print_emoji_detection_script' );
    remove_action( 'wp_print_styles', 'print_emoji_styles' );
    remove_action( 'admin_print_styles', 'print_emoji_styles' );    
    remove_filter( 'the_content_feed', 'wp_staticize_emoji' );
    remove_filter( 'comment_text_rss', 'wp_staticize_emoji' );  
    remove_filter( 'wp_mail', 'wp_staticize_emoji_for_email' );
    add_filter( 'tiny_mce_plugins', 'disable_emojis_wp_tinymce' );
}
add_action( 'init', 'disable_emojis_wp_head' );

function disable_emojis_wp_tinymce( $plugins ) {
    if ( is_array( $plugins ) ) {
        return array_diff( $plugins, array( 'wpemoji' ) );
    } else {
        return array();
    }
}

// Remove other crap in your example
add_action( 'get_header', function() {
    remove_action('wp_head', 'rsd_link'); // Really Simple Discovery service endpoint, EditURI link
    remove_action('wp_head', 'wp_generator'); // XHTML generator that is generated on the wp_head hook, WP version
    remove_action('wp_head', 'feed_links', 2); // Display the links to the general feeds: Post and Comment Feed
    remove_action('wp_head', 'index_rel_link'); // index link
    remove_action('wp_head', 'wlwmanifest_link'); // Display the link to the Windows Live Writer manifest file.
    remove_action('wp_head', 'feed_links_extra', 3); // Display the links to the extra feeds such as category feeds
    remove_action('wp_head', 'start_post_rel_link', 10, 0); // start link
    remove_action('wp_head', 'parent_post_rel_link', 10, 0); // prev link
    remove_action('wp_head', 'adjacent_posts_rel_link', 10, 0); // relational links 4 the posts adjacent 2 the currentpost
    remove_action('template_redirect', 'wp_shortlink_header', 11);  
    remove_action('wp_head', 'adjacent_posts_rel_link_wp_head', 10, 0 );
}, 99);

// Remove adminbar inline css on frontend
function removeinline_adminbar_css_frontend() {
    if ( has_filter( 'wp_head', '_admin_bar_bump_cb' ) ){
        remove_filter( 'wp_head', '_admin_bar_bump_cb' );
    }
}
add_filter( 'wp_head', 'removeinline_adminbar_css_frontend', 1 );

And if you want to remove the rest api link I believe this will work:

remove_action( 'template_redirect', 'rest_output_link_header', 11, 0 );

See this WPSE post here for additional solutions removing rest api.