This gets tricky for several reasons: The current law on the matter is largely dictated by the Vienna Convention on Diplimatic Relations, 1961. He was born in 1961. While the US and Iran both
signed the Vienna Convention in 1961, the US ratified in 1972, and Iran in 1965.
Current US law on the matter (22 USC § 254) is pretty much entirely defined in terms of the Vienna convention. As such, current US law on the matter is pretty irrelevant.
I have no idea what the legal situation was when he was born. Likely there was still a distinction, but even today the Vienna Convention, to the extent one can use it to guess whether it's believable that he might be affected (and it's perfectly possible the US State Department official who caused this is just wrong), in article 37 grants "members of the administrative and technical staff of the mission, together with members of their families forming part of their respective households" who "are not nationals of or permanently resident in the receiving State" limited immunity.
So you're right they would not have full immunity under the terms of the Vienna Convention, but some immunity does extend to family, and the question then becomes how much immunity did they have under pre-Vienna Convention US law, and how much immunity is sufficient to justify legally withholding citizenship.
My best guess is that this guy will get his citizenship back, and that it will turn out some overzealous bureaucrat saw that the dad was embassy staff, saw the word immunity, saw the word immunity on some checklist and didn't check the details - whether or not there'd be some technical possible justification for withholding it or not. Hopefully for this guy someone decides it's a stupid waste of both money and political capital to pursue this before he ends up spending even more on lawyers...