I love this conversation, and I'd like to give my feedback. First, no, you are coming across as a person who thought deeply about this concepts.
I like the idea of closeness but, as you noted, it is not always possible. I think that empathy is the right word, but should be better characterized. Empathy is trying to establish a connection, is to bring the two people close when giving the honest feedback so that it is perceived correctly. How to do it, that's another story.
One technique that I found very powerful is called feedforward. The idea is, instead of saying what someone has done wrong, suggest a way he could make it right the next time. Example, you sent an email without the attachment. I could tell you: "you should be more careful!" or I could give you a constructive idea, maybe creating a connection: "I understand that sometimes it happens to forget an attachment. I developed the habit of putting the attachments in the email before writing it, so I don't forget them". It is not a perfect example. But even if we aren't close, it gives you a sense of closeness, a feeling that I understand you, and I care about you.
The problem with this approach is that the receiving party should be receptive. I used this method and I received answers like "But I believe I was right because...". And this is the end. You could ignore the answer, knowing that your feedback was useless, or you could try to explain, but you have lost. Nevertheless, if the feedforward is ingrained in the company culture, it is something known and actively practiced (that means the example starts from the CEO), it helps to create a positive environment.
I'm not sure if that's what he means by empathy, or maybe his company uses a different technique. You are right: closeness, privacy, and maturity are essential for a positive culture were front-stabbing (or worse, backstabbing) is impossible. And empathy is the ability to bring someone close, especially when it is relevant to provide important feedbacks.