It's not entirely impossible. There are newly appointed VPs that are not scholars or FLP. However, they would have met the right people at the right time in their careers, and demonstrated that they are ready for the job. Often they are identified early enough to catch the boat, getting job rotations along the way to diversify their experiences.

If you are not earmarked for an "automatic" career path towards it, you need to create the path yourself. Move environments to find good leaders who sync with you, apply to HQ for exposure beyond school context, find your own mentors in the fraternity and build connections / networking. Actually it's like any other job in the world if you want to move up.