My ds7 is in public school (1st grade). We too live in an area with a respected university. The problems in for him in elementary are: he's not very "mathy", the 1 or 2 kids I've seen in the past 2 years were mathy girls. Both years, he's the only boy in his group. Essentially, he feels like he doesn't have peers.
I know other kids/ people have good experiences at public school. Just thought I'd share our experience. I will say things seem to greatly improve by middle school. We'll see.
Good luck. Remember if you try something and it doesn't work, try something else. It's hard to know what will be a good fit for the child without trying.