Scores help admissions officers determine if you can handle the academic work.
To be admitted, you need to demonstrate that you are likely to make a positive and meaningful contribution to the community. That’s where your character and personality come in (as demonstrated in your essays and recommendations) and your extra curricular activities.
Also: Try not to think in terms of “top” schools. What does that even mean? The teachers are comparable at most boarding schools. The best school for you is the one where you thrive and do your best work… and where you get the most generous financial aid package.