How do I make a draft post accessible to everyone?

You cannot assign capabilities to unknown users. If you want to make a post visible for everyone, create a separate URL for these posts and add a control element to the post editor to enable the preview on selected posts only.
When such an URL is called, check if a preview is allowed for the post and if the post hasn’t been published already. Also make sure search engines ignore this URL.

For the URL I would use an endpoint:

add_rewrite_endpoint( 'post-preview', EP_ROOT );

Now you can create URLs like …

http://example.com/post-preview/123

… where 123 ist the post ID.

Then use a callback handler to inspect the post ID, check if it is valid and overwrite the main query. This is probably the only acceptable use case for query_posts(). 🙂

Let’s say the endpoint is a class T5_Endpoint (a model), and the output handler is a class T5_Render_Endpoint (a view) which gets the model passed earlier. Then there is probably a method render() called on template_redirect:

public function render()
{
    $post_id = $this->endpoint->get_value();

    if ( ! $post_id )
        return;

    if ( 1 !== $this->meta->get_value( $post_id )
        or 'publish' === get_post_status( $post_id )
        )
    {
        wp_redirect( get_permalink( $post_id ) );
        exit;
    }

    $query = array (
        'suppress_filters' => TRUE,
        'p'                => $post_id,
        'post_type'        => 'any'
    );

    query_posts( $query );

    add_action( 'wp_head', 'wp_no_robots' );
}

$this->meta is another model (class T5_Post_Meta) for the post meta value that controls if a preview is allowed. The control is set into the Publish box (action post_submitbox_misc_actions), rendered by another view that gets the same meta class.

So T5_Post_Meta knows where and when to store the meta value, the views do something with it.
Also, hook into transition_post_status to delete the post meta field when the post is published. We don’t want to waste resources, right?

This is just an outline. There are many details to cover … I have written a small plugin that shows how to implement this: T5 Public Preview.