The missing ingredient here is Google punishing bad notification categorization in some way (e.g. by diminished ranking in the Play Store, or failing the review process), but seeing how Google itself doesn't isn't too keen on clear categorization in their own apps, I'm not holding out much hope.
I particularly miss the times when Facebook would send notifications to your e-mail, with full content included, within minutes to hours of an event, if you happened to be logged out from the service. This was a neat way of reading chat messages without making the sender aware you've read them (that was before the whole thing got separated out to become Messenger).
EDIT: one major benefit of e-mail as notification channel is that a typical non-ISP personal mailbox has practically infinite storage and a sensible search mechanism, giving you historizing/archiving for free.