Extend WordPress Core Classes in OOP Theme?

I am trying to use $wp_admin_bar to remove a couple of the default
WordPress designs but I am not able to find where to add $wp_admin_bar
that does not trigger an error.

The wp_before_admin_bar_render hook doesn’t pass the admin bar object, but you can access it manually (from within a function or a class method) using global like so, just like the Codex example here:

function remove_wp_logo() {
    global $wp_admin_bar;

    $wp_admin_bar->remove_menu( 'wp-logo' );
}