The only problem with that idea is this one:

1. Send something to the e-ink.
2. Battery dies before you finish what you're doing and the phone shuts off.
3. No battery/phone shutdown means the e-ink never appears and you lost the benefit of being able to save something for when the phone's dead.

Surmountable problem (and probably a rare issue anyway) but would be irritating if you were in that situation where you don't realise your battery is so low so your phone shuts down before turning on the e-ink!