Yes, lots of top private colleges would provide a full tuition scholarship and some financial aid towards room/board - there’s been an initiative to guarantee at least full tuition to families that make up to 200k.
(Not all, of course - and public universities use merit, not need, and prioritize instate status, etc.)
Run the NPC (net price calculator) on Rice, Emory, Vanderbilt, Davidson, Amherst College, Brown, U Rochester, Cornell. Which 3 have the lowest net cost for you and what’s that net cost?