Sure it's mystical. You know what you mean by connection, but surely you don't believe that everyone shares your understanding of it. Do you really feel that someone with no prior experience, walking into the dojo, has your understanding of the word "connection" to mean "a function of proper body structure, movement and mechanics"? To that person, simply talking about "connection" without explaining what you mean by that term (using the words of a common language together with demonstration and practice), is absolutely mystical.
No, that's why when I teach I spend the time trying to explain the notion in detail as part of what I'm getting at rather than just leaving it at the word. That's why I will often start with so-called "ki exercises" then try to explain what I think is going on in terms that they might be able to understand. And not just settle for "let your ki flow". Then I will move on to a technique that I think instantiates that particular skill and hopefully get them to better understand what I'm getting at. Connection is a common underlying theme in most of the classes I've taught lately. I'm a little baffled as to how you can tell me how I've been using the word in my classes.
So getting back to my original comment, I try to avoid saying "don't try to throw" because I think it is often misunderstood. I will instead talk about trying to use the entire body as a whole and work on their basics of movement. Then if it does come down to that I already have a context within which to explain what I mean.