Sure, but that's ignoring the cost of "now your users don't trust the security team".
For most things like phishing there's only so much training you can put on a user. Humans are pretty okay at understanding the costs associated with their time in an implicit manner. Users will check well enough to meet their internal cost metric: the cost to them if they get phished isn't high, and the likelihood is low. That's why it's such a problem in workplaces.
The solution isn't to keep beating the user over the head. First, it can undermine other important parts of the relationship between users and security as I mentioned, and it can , if done in the extreme, normalize phishing emails. The real Phish comes in and sits unreported next to the fake ones. Security never gets to run a scan and remove the message from every mailbox, increasing the exposure.
The better approach is to prevent users from being in control of their own vulnerability. Don't let them enter their credentials into the nono box.