No response from Journal editor after misinformed rejection
I would personally start wrapping my head around the idea that you will (a) not get the "clear and concise" explanation from the Editor-in-Chief that you are hoping for, and that (b) this publication is unlikely to appear in this specific journal. In short, it may be time to move on and think about alternative publication venues.

On this site, we have many questions related to appealing journal decisions, and (in cross-field agreement that is rather rare around here) most experienced researchers here agree that fighting a review decision that you perceive to be unfair is a wasted effort. Decisions actually being sent out "by mistake" is very rare (to the point that I am not aware of a single such case in my circles), and that the authors do not agree with the content of a review is basically never grounds for a re-evaluation of the manuscript.

The fact that the editor is not really getting back to you on this is another strong indicator that this is not a case of a silly mistake, but that the journal stands behind their decision and wants to cut short any possible discussion related to this manuscript. 

You can definitely draw your own conclusions from this experience (such as to not submit to this journal again), but there is no formal recourse against the decision of a journal to reject a paper, and them backtracking on this is unlikely enough that you should not delay your manuscript further while you are trying to fight.