Hi Trinko,
You have no reason to worry:
1. Both Kellogg and Yale ask for the career goals in work experience sections of the applications. Albeit, they require very short descriptions yet the questions are very much there. Kellogg needs about a line for easy short answer whereas Yale asks for a 250 word essay.
2. On the essay front- Yale has one essay on how you impacted the organization where you work. Your organization is the school and just like any organization has clients, your school's clients are its students. As a teacher you impact their lives and these would be your achievements. You could have initiated some processes in class through your teaching, or improved the school curriculum, or just touched the lives of some students in a big way - all of this is your impact. Use this essay to your passion for education and this will tie in nicely with the short career goals essay.
For Kellogg- you have two essays which sound very relevant to your work experience. As a teacher I can imagine you facing many challenging situations that require perseverance and grit (essay 1) and also being in leadership situations where you motivate and collaborate. Motivate- that's a teacher's primary job, right?!
Both these b-schools also have a video essay component and Kellogg requires you to interview as part of the application process.
So I see a lot of opportunities for you to make your story heard. Cheer up and get on the task.
Both schools are going to view your social impact highly. So you need not worry about not working with a Fortune 500.