Suggested by Frédéric Vercaemst – Completed
In the webclient, page Design is limited to adding/removing field controls only. The other actions (Part / Page / Action) are still disabled. This prevents users from adding factboxes that are not visible by default. Please enable full customisation features as available in the current Windows client (before killing it).
Status Details
Good news! We released the ability to bring back factboxes as part of the functionality in Unhide Parts on a Page.Please refer to the following link for more information:
Best regards,
Mike Borg Cardona
Program Manager, Microsoft

Indeed, this is inconsistent. It is also not possible to adjust the actions that can appear on factboxes through customization, which is something we're really missing since leaving the RTC. We should be able to surface and hide factboxes at-will.