Can’t publish custom post type – “You are not allowed to edit this post.”

The default for map_meta_cap is actually not false if you’re also passing in a capability_type of post or page, which you are.

The following code is in WordPress Core’s post.php:

// Back compat with quirky handling in version 3.0. #14122
if ( empty( $args->capabilities ) && null === $args->map_meta_cap && in_array( $args->capability_type, array( 'post', 'page' ) ) )
    $args->map_meta_cap = true;

So by explicitly setting it to false, you were telling it not to use the capabilities of a post or page.

By not setting it at all, you allowed the above code to set it to true.

Code just a few lines down, get_post_type_capabilities uses this flag for whether or not to use the default posts capabilities which gives you the permissions you’re looking for.