Every email that I have received from a legitimate organization in the past several years has had a one- or two-click unsubscribe link at the bottom. I can't get access to the Red Cross' reminder emails right now, but given what the Red Cross is, it's rather likely that the parent is telling the truth.
Why the minuses?
I don't share my email all over the web, I usually don't sign up to many websites. If I do in majority of cases (99% I would say) there is included option to manage my subscriptions and/or unsubscribe directly from the message... and it works.
Many literally have a link at the bottom that says "unsubscribe". I've received emails from dozens of different services and companies over the past few years and every single one, without exception, has had this feature. It's prevalent enough that my email client even attempts to find it and presents me with a button in the UI if it does.
Notably CAN-SPAM act requires spam to have effective and visible unsubscribe links. While this doesn't impact off-brand Viagra and Nigerian prince spammers, most legitimate US businesses (such as American Red Cross) should be compliant.
Edit: parent