They all manage the data for their services. Drafts does not, it is held by Apple’s iCloud platform. Without you being authenticated to something (like your Mac) to do the data shunting, you would need to have something that authenticates you to iCloud and writes directly to your Drafts database.
Maybe it’ll come one day, but Greg would need to develop a whole web-back end platform to support this sort of thing.
As far as I know, none of the usual web services support iCloud Drive, so that leaves you needing something like a Mac and Hazel as I originally suggested to go the file import front.
Another way that is roughly equivalent, and what I currently use, is the even older Apple Reminders import functionality. It works the same way as the files import, but takes the draft content from a reminders list.
I login to iCloud on the web and add my content to reminders directly, but I’ve just done a few checks and I was able to replicate the process end-to-end by setting up an IFTTT applet.
The applet is set to receive e-mails (from my registered e-mail address) tagged
#drafts in the subject line and add the body of the mail to my Drafts import list in reminders. Then, opening up Drafts on the device I have set to do the import, the e-mailed in information is imported and appears. I wouldn’t be surprised if there was a size limit, but I’ve not encountered it in my day-to-day use as yet.
Hope that helps.