add_menu_page does not render the expected result

as the 5 parameter to add_menu_page() you send the resultat of running the function, insted of sending a callable reference to the function, as you did in the add_action.

$hookname = add_menu_page (
        'My Slider',
        'My Slider',
        'manage_options',
        'my_slider',
        array( 'My_Slider', 'render_admin_menu')
    );