It just occurred to me that you probably have another issue in mind as well.
Is the child subject to US jurisdiction? Yes...unless the Saudi ambassador married the mother and did whatever paperwork is necessary to have the child added to his household. I don’t know what that would entail. I also don’t know what rules the State Department uses with the families of ambassadors if they get into legal trouble. I imagine this would be a sticky issue if a member of an ambassador’s household was also an American citizen. If I were making the decision, I would insist within the paperwork that the child be removed to KSA during the time that the ambassador was posted to the US. But I don’t know what would actually happen.
Yes, but so is the ambassador. That issue was also settled in the senate debate for the Citizenship Clause. An ambassador can't murder a US citizen and be immune from prosecution. At least, that was the understanding of the Framers of the C.C., and that was one of the reasons they decided that the words "and subject to the jurisdiction thereof" would so obviously be taken to mean “full and complete jurisdiction” that any further clarifying language was totally unnecessary.
I imagine this would be a sticky issue if a member of an ambassador’s household was also an American citizen.
That's just it, there wouldn't be a child born into citizenship in that household. To me, it would seem impossible to even naturalize the child until the child was old enough to renounce all possible conflicting allegiances.
But I don’t know what would actually happen
Your problem is that you aren't sure how some aristocrat appointee of the King of England would have adjudicated the matter 500 years ago. It's clear what the Framers of the clause had in mind.