This file only handle shipping/billing address fields display.
The customer email is not an address field and there is no tag available for it on that view.
It's actually not that view that you should edit but the view files which call that address_template view file.
For example, for the invoice, it's the "invoice" file of the view "order" for your backend template where you can add:
echo '<br/>'.$this->order->customer->user_email;
after:
echo str_replace("\n","<br/>\n",trim(str_replace("\n\n","\n",preg_replace('#{(?:(?!}).)*}#i','',$html)),"\n"));