Solution: Website toolbar for custom classes

By default, the website toolbar in 3.9 is only shown for classes shipped with the website interface. If you want it to be shown for a custom class, add your class' name to the AvailableForClasses[] array in extension/ezwebin/settings/websitetoolbar.ini and clear the cache ;)

If your class is a container, you also need to add it to the ContentClassContainers array in the same file to enable creating objects under it.