Why does hook priority affects admin menu permission error?

Priority

All hooks & filters have a priority. Those are used to order the functions that are attached to hooks. By default the priority is 10.

As the default is 10, there’s a big chance that your filter/action gets hooked too early and therefore can’t affect the actual output.

Edit:

1. Imagine that the plugin hooks it’s bootstrap into the 'admin_init' hook to add something that needs permissions to access. Inside that bootstrap it hooks something to the (fictional) 'fantastic_hook'.
2. Imagine that you hook some extension of a plugin to the 'after_theme_setup' hook. There you add some function to the (fictional) 'fantastic_hook' too.

So the result of print_r( $GLOBALS['wp_filter']['fantastic_hook'] ); would be the following:

[10] Array => ['fantastic_hook']
    ( 
        Array => ['your_custom_extension']
        Array => ['the_plugin_bootstrap']
    ) 

As your extension and the bootstrap run both at priority 10. But yours got added first and will therefore run first. And there comes the problem…

About the menu items

The admin menu works pretty similar: You have a priority (the (sub-)array index) that cares about ordering and placing the menu items.

// Example for your functions.php:
function dump_filters()
{
    $output="";
    foreach ( array( $GLOBALS['menu'], $GLOBALS['submenu'] ) as $item )
        $output .= var_export( $item, false );
    echo "<pre>{$output}</pre>";
}
add_action( 'adminmenu', 'dump_filters', 999 );

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Note: I wrote a pretty intense post about the menu order, where you can see how you’d reorder the menu and read a little bit about the behind-the-scenes of the menu and it’s limitations.