Except you were emailing to Amazon. You can actually have a real email Client on an Android eink (not Google's Android Mail, which is just giving your credentials to Google's servers), like K9 Email. I've not tried emailing an epub or PDF to it, but I presume I can. Unfortunately even with KOReader I'd rather manage ebooks on the Kobo with Calibre and browse 5000 ebooks by metadata, not filesystem.
I must try a direct email

Which no Kindle has ever had.