The child-html of (pretty much) any block can be found by looking through layout XML. It can also be added/modified through the
Block classes that you mentioned, but that doesn't help with this case, so I digress.
The actual location of the file may vary based on your theme and if you have overridden the default settings, but for the magento-luma theme, the file is located under
magento/theme-frontend-luma/Magento_Theme/layout/default.xml, where you will find
<block class="Magento\Framework\View\Element\Html\Links" name="header.links"> on line 11.
For many blocks, the output is controlled through the template (.phtml) file that would be defined in this line, but seeing as no template is assigned, the output generated is controlled through the
Magento\Framework\View\Element\Html\Links class, found in
vendor/magento/framework/View/Element/Html/Links.php. I hope this helps!