No rejection letter?

Then they didn't generate a letter to protect themselves. Letters constitute proof and the letter would have had to give a reason.
Having taught at a private school for many years, I'm aware of the back-handed way they can work in and I sincerely hope this works out for you.

Oh, I don't think a concern about appearing "oddly cold" had anything to do with not receiving the letter. They were oddly cold throughout the whole process. I thinkthat refusing my son based solely on his disability gets into a hazy area since tuition remission is a lost employee benefit worth $300,000 and is viewed, rightly or wrongly, as an entitlement. My son is the only faculty child not to be accepted according to staff. He has been accepted by another competitive private school, so one offer that we would have appreciated would have been some assistance with tuition, just as if we had refused another employee benefit, health insurance, and been given some reimbursement. I highly doubt that is going to happen, however. And if discrimination is perfectly legal, could a private school discriminate on the basis of race? I think not--at the very least, they would be risking their tax exempt status. My son's academic performance is not affected by his disability, as several tests have indicated. It is prejudice, and the school psychologist has told us as much and advised us not to discuss his disability because the school faculty are too ignorant about it.

Private schools are under no lawful obligation to accept any child and can refuse a child for most any reason. As your husband's on the faculty and closely connected to the school, he can certainly ask his colleague why your son was not accepted. When they called to tell you this, what questions did you ask then?
You likely received no rejection letter because to send a letter to faculty member would be oddly cold- at the least you deserve a phone call but your husband really deserves a personal explanation.

It's your husband's privilege as a faculty member to speak with the admissions people or the headmaster. It may indeed be the case that your son is being denied because of a mild learning disability but there's no reason they shouldn't be able to say that to you directly.

Heloise, my son is also gifted. There is a term for gifted children with learning and/or attention difficulties, and it's "2e" or "twice exceptional." If you would like to learn more, I invite you to look at and join the "2e Group" at community.greatschools.net/groups/16042