@pichalite that was it. It's a bit confusing... Maybe the private option for specific groups should be grayed out if the general option for allowing private groups is disabled, to make this clearer.
Can't hyperlink to API pages.
Clicking this link will cause a 404 error, hard refresh the page and it loads fine. May be worth checking for siteURL/api/ and opening that in a new tab or something as a workaround.