Ms. Future Education Administration

You’re no slouch – solid GRE score, Swarthmore UG (and difficult STEM major at that with a 3.7 GPA!). Your professional interests have impact on humanity (yay education!) rather than simply shareholder value, and that will also resonate with the AdComs.
GSB and HBS have strong dual degree cultures and the HBS MBA + Harvard Graduate School of Education (HGSE) Ed. M is an especially powerful combination for your professional goals.
Your professional strategic positioning (e.g. what you want to do post-MBA and why) is unique and seems genuine. It syncs with your prior professional experiences. GSB will believe that you want to do these things (which is positive, as oftentimes “believability” of strategic positioning, especially in social impact-type routes like education, is a major hurdle). I’m a bit worried, though, that the AdCom will wonder why you need a GSB MBA to achieve your goals. The product management in edtech – sure. That’s right up the fairway of post-MBA bay-area employment. But education administration doesn’t really make sense and the flow of career post-MBA doesn’t naturally hang together. Education administration, which makes sense for you because it’s already in your background, doesn’t naturally flow from product management in edtech and typically doesn’t require a GSB MBA. Depending on the role you are seeking, I’m also worried that the long-term goal of K-12 Head of School isn’t going to seem ambitious enough for the AdCom. CEO of a large charter school network with great metrics of success for underserved children? Now you are cooking with gas. BUT, this sort of administrative role ALSO doesn’t really flow with edtech product management/product strategy. It makes sense that you want to be in education, but your short-term, mid-term and long-term goals need to hang together and present a plausible progression that is both imminently achievable and GSB-level ambitious.
Given your excellent stats and assuming flawlessly executed application and watertight strategic positioning (which will require ironing out some of your career plans), I’d estimate your probability of success with GSB to be triple the average 5% GSB acceptance rate, about 15%. That’s a shot worth taking and we are rooting for you!