(Also I wonder if using the word "signature" only makes it harder to explain. Like, trying to explain to someone how an analog signature proves your identity. Because, well, it doesn't.)
It also neatly explains the concept of key sharing, which seems to present some pedagogical difficulty for some reason - you can sign your letters all you want, but if I've never seen your signature before then it doesn't actually prove anything.
Public key = knowing what your signature looks like
Private key = the muscle memory required to create a signature
By now every single email user with an email address that has been exposed to the internet has received untold amounts of spam emails with their own address spoofed as the sender. The only question is if they've ever noticed it.
This wouldn't pass most spam filters today, thanks to stuff like DKIM: https://en.wikipedia.org/wiki/DomainKeys_Identified_Mail
But I agree that this is rather intransparent for the user. What we need is a warning like "You have never received an email from this domain. Do you want to trust it?" and something like the Extended Validation Certificate for companies (does something like this already exist?). Not PGP though.
But they are right in one sense: it's their mail provider's job to provide the signing infrastructure.
But customers have to know they need it and demand it
There, conceptually it's explained, to anyone who has any concept of what "writing" is.