Even if it’s a school merit (as you mentioned: Presidential scholarship), that will get applied FIRST to your need. Then if you still have need, then a school may add some need based aid to cover the rest of your need.
RARELY, if ever, will a school apply need based aid to cover “need” and then apply their own merit in addition to cover some of your EFC.
If you want to reduce your EFC, you need to target schools where the merit is so huge, that your net cost is what you want.
COA doesn’t change…it’s the listed cost of the college. It’s not your net cost.