You give yourself other options - and frankly better odds - to sub in the high pedigree LACs with schools like Union and Trinity (say a match) and Clarkson (a safety, albeit regional) - fine LACs that offer ABET accredited engineering programs.
Obviously, CS programs are offered in more places of course - and all schools have pre-med.
We don’t know the school rank - but in general, a 3.66 UW will be a reach for much of the schools.
Since you prefer private and are ok with ED, you might look at mid size schools like Rochester and Miami as well - these are difficult admits but not the top.
If you’re going to med school, there where won’t matter. For engineering, the where likely won’t matter but he’d get into plenty of wonderful schools - small like you laid out or mid size and large.
And if he wants to study CS, it’s at so many places - and many today require you to pass a series of tests to get hired - regardless of the where. My nephew studied Poli Sci at Arizona and works for a large, well known company in NY. He went in as a mid hire - but had to pass all the tests.
Bottom line -with your student, there is no shortage of possiblities.
However, do we know what your student wants. You prefer small and private but then are talking about UCs which are large and bureacratic.
I would take your son around to different schools - schools that are urban, suburban, rural. Large, mid size and small.
How does he feel about cold weather, etc.?
All these things - I’d first focus on, what does he seek in a school and build up from there.
Rutgers is different than a UC which is different than suburban school like Wesleyan and which different than a school in Iowa like Grinnell, which is in the midst of an area without a huge Asian population. And would that be comfortable.
So that’s, to me, first step - and then of course, if engineering is a possibility, only visiting schools where it’s offered.
Good luck.