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So your question is "What score should I aim for?" and I think the answer to that should always be "as high a score as possible". But we can expand your question a bit - how high of a score do you need to potentially get in to *a* school, and how high of a score do you need to get in to a specific school?
If we look at the overall grid of MCAT and GPAs, we can see that among those with a total GPA 3.60 - 3.79 (and you're at the tippy top of this group) that the acceptance rate cracks >50% at the MCAT 506-509 category. Of course, it goes up to 66% at 510-513, 75% at 514-517, and 83.3% at >517. In addition, that grid includes all comers - including those with activities that are significantly less impressive than yours. We know that many medical schools like veterans, and you seem to have hit all thje right points to have a compelling story.
In addition, we know that ethnic minorities deemed "underrepresented in medicine" (URM) are a focus for a number of schools, with the mean African American matriculant having an MCAT score of 505 compared to the overall average of 511. Does this mean being African American is worth 6 points on the MCAT? I think that would be a statistical misunderstanding (particularly given the existence of the 3 historically Black schools), but we know that it is overall a plus to your application. They used to put out grids by specific race, but haven't done that since 2016. If you look at the one from then, an African American applicant with a GPA in your range needed an MCAT in the 24-26 range (roughly 498-502 on the new MCAT) to have a 75% chance of getting in.
Finally, we have to discuss your specific schools - they include some of the most competitive institutions in the country. Stanford has an average incoming MCAT of 518. UCSF 516. NYU? 520, and thats before they made school free. Just glancing at your list, the lowest competitiveness school is probably FIU (I don't know anything about it), with an average MCAT score of 510. Now, do you need to be above the average score for a matriculated student at a particular school to apply there? Well, obviously not, because half of their matriculants are below that score. But being in the ballpark is likely helpful.
So what I would say is that a score in the 506-509 range (given the impressiveness of your ECs and your ethnic background) will lead you to a an overwhelmingly likely chance of getting in somewhere if you apply broadly. Less than that and you'll still probably get in, but the lower it goes the worse your chance is. To get in to NYU, even with as compelling an application as you have, my bet is you'd need a score closer to the mid 510s.
Of course, I applied to medical school 10 years ago, so probably one of the pre-MD experts can give better advice here - I really just got drawn by the mention of shadowing an endocrinologist 😉
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