I think I may have misread your original post. Do you have your employee contacts setup with two different anniversary dates - one that's an actual anniversary field that the automation is based off of, and the other a custom field that will actually be set to display written out?
If so, then you can completely ignore my original response. When you preview your email or send a test, the contact detail displays the personal information you provided for your own account to give you an idea of what your contacts will see. If the information isn't available to pull into your preview, or if you're using custom fields, the fallback text displays instead. As long as you've gone through the insert contact details process, then it will trigger as expected. If you just manually typed out [[custom.anniversary_date OR ""]], then it won't trigger properly since it's not coded.
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