Thanks to the answer of @klaussner over here, I might have found the answer to get all content in a space.
The v1 descendant
endpoint (not deprecated!) can also handle the new content types (although the docs don’t list them…).
So starting with the root content(s) of a space, we can call
/rest/api/content/<ROOTCONTENTID>/descendant/whiteboard
/rest/api/content/<ROOTCONTENTID>/descendant/database
/rest/api/content/<ROOTCONTENTID>/descendant/folder
This collects everything for the space.
(Not sure yet, though, how to get all root contents for the space if those are non-page.)
Edit: Even better, we can use rest/api/content/<ROOTCONTENTID>/descendant?expand=whiteboard,database,folder,embed,page,blogpost
to get all those. This is documented.