Hello bgbtech, thanks for your question.
Regarding your GMAT quantitative score, another way to mitigate it would be to cite an analogue that is a better representation of your quantitative ability and potential. Programs would want to know that you can handle the coursework.
Your competitiveness will really come down to the quality of your work experience. Can you draw upon “moments” when you truly led: influenced individuals and organizations, drove results, took initiative?
In terms of your career “story,” the onus will be on you to paint a picture that cohesively ties together your past, present, and future and demonstrate to the admissions committees that your desired career path necessitates an MBA. How does product management connect with your work and extracurricular activities? You will have to compelling answer this question to be competitive.
I think your school categories are generally fine, but I think Northwestern and Chicago might be slight stretches too. Ultimately, it really depends on whether you can mitigate the quantitative GMAT score (by retaking the GMAT or otherwise), how well you call out your leadership experience, and if you can connect your past to your post-MBA goals.
Best of luck!
Brian
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