I must not be searching for the correct thing as I fail to find even one service doing that.
Any advice?
User visits your site and wants to upload a document. They see the logos for Box, Dropbox, Drive, and click on whichever they prefer/have an account for. You authenticate then upload the document to a specific folder in a specific namespace in that drive (user_drive_box_dropbox/acme_saas/foo.pdf).
For people who don't have an account in any, you can use the same integration to upload documents to your drive's acme_saas_drive/user_id/foo.pdf.
This way users have control over their data, and you fan out adding integrations one after the other.