Hiding your employer would not appear to be of any use at all when you want to hide the employee's email address from the public. If you hide your employer info but spread your contact details far and wide, the employer info is not interesting.
The assumption being made is that once you know the company name and the employee name, then one can freely email the employee. Trying to address the threat of incoming emails by trying to hide the company name, so that the email address domain can't be guessed, so that emails cannot be addressed is trying to push on the wrong end of the lever of control. And you are trying to do it with a wildly difficult policy to enforce.
The trivially effective control is to break the direct tie between company name, employee name, and email address.
I know of companies that stand up a separate domain to send emails from. So
example.com stands up
example-email.com. This immediately wipes out a lot of automated emails. Other companies salt the email address with 2-4 numbers, so
email@example.com. Others use only the employeeID number:
While each one of these can be overcome through analysis of other disclosed email addresses from the company, it is more effective and much, much easier to control and enforce through technical means than forcing people not to disclose where they work.
The company name is simply not the primary data to control in this threat scenario. It's the email addresses. You can control those.
Managing digital footprint is always a good consideration but you have an awareness problem and a trust problem with your employees that such a policy is not going to address.