wp_insert_post to schedule a post – but nothing happens?

As Rarst mentioned I had to use wp_update_post(), but then there is a little trick – one must also set edit_date = true otherwise it has funny behavior. The final code looks like this:

function schedule() {

$postdate = date('2014-06-11 01:00:00');
$postdate_gmt = date('2014-06-11 05:00:00');
$post = array(
    'ID' => 11,
  'post_status'    => 'future',
  'post_type'      => 'post',
  'post_author'    => '1',
  'ping_status'    => 'closed',
  'to_ping'        => 'http://rpc.pingomatic.com/',
  'post_date_gmt'  => $postdate_gmt,
  'post_date'  => $postdate,
  'edit_date' => 'true'
);  

wp_update_post( $post, true ); 
}

add_action('wp_head', 'schedule');

And here’s how it can work for many posts, using a text file:

Text File:

postid,server_time,gmt_time
postid,server_time,gmt_time
postid,server_time,gmt_time
postid,server_time,gmt_time
...

Function:

function schedule() {

    $fh = @fopen( dirname( __FILE__ ) . '/schedule.txt', 'r' );

    if ( $fh ) {
        while ( ( $line = fgets( $fh ) ) !== false ) {
            $ids = explode( ',', $line );
            array_walk( $ids, 'trim' );

$postdate = date($ids[1]);
$postdate_gmt = date($ids[2]);
$post = array(
    'ID' => $ids[0],
  'post_status'    => 'future',
  'post_type'      => 'post',
  'post_author'    => '5',
  'ping_status'    => 'closed',
  'to_ping'        => 'http://rpc.pingomatic.com/',
  'post_date_gmt'  => $postdate_gmt,
  'post_date'  => $postdate,
  'edit_date' => 'true'
);  

wp_update_post( $post, true ); 
        }
    }

    }

add_action('wp_head', 'schedule');

Many thanks to everyone who helped in this and other threads!