Both intelligence and emotion is an indispensable part of the educator and teacher. I think the two concepts put together help to understand the fact that emotional intelligence is a presentation of the theory and practice put together. In the figure of the teacher it is essential to work both on the theoretical level (all training and self-training courses), and on the operational level, that is, in the relationship with other people.
Thanks for you interesting question. What follows is a short response to it. I do consider emotional intelligence a need in teacher's training. For many and many years, primary school teacher's training, for example, was focused on his/her competence to teach children the three "Rs" of cognitive intelligence (wRiting, Reading, Reasoning). Nowadays a comprehensive teacher's training is also focused on the three Cs of emotional intelligence (Care, Concern, Connection).
Intelligence emotional is important in a teacher, I am a psichology and I had six years working in Education in the university where I learned that a teacher of any line teaches better and achieves that his students find the pleasure to him to what they do thanks to the emotional state that the teacher puts him. There are theories that support emotional intelligence in teaching and it is: "The pedagogy of tenderness or also known as the pedagogy of love". In summary, teaching based on emotional intelligence generates the student learning with the will and desire to want to do things or to do things because they have to.