What you are suggesting is actually quite complex. Mailchimp would have to install a bunch of stuff on customers servers (which are probably run by a third party) and/or control some of their customers subdomains.
Maybe you just need to invest some money in better spam filters, and user education.
This year the Canada Revenue Service is running an ad campaign saying they aren't calling/emailing Canadians and demanding they go buy gift cards at Rexall/Shoppers Drug Mart/Sobeys in order to not go to jail for tax fraud.
I'm sorry but there is no level of user education that will protect a large organisation from phishing links. The attack surface is way too large and safety depends on how the user is feeling on any given day when a random email turns up. There are no high reliability systems that require manual user interaction on a frequent basis especially on low-effort low-concentration tasks.
I agree about the insufficiency of user-education.
I think we need go further: what we need to do is kill the idea that email is an good vector for notifications inside of big organizations (or maybe even outside of them). I should have a dedicated app with a whitelist based system that official notifications go through. This could be not only a security improvement, but also a stab at better UX, better productivity, and avoiding notification fatigue.
Maybe you just need to invest some money in better spam filters, and user education.