Actually, BA only fly thrice weekly, or about that, to RUH.
What Jr said about entry requirements was correct though.
The problem with flights from the US, is that the traffic would be highly seasonal and nothing else. Yes, there are large concentrations of foreigners, including US citizens, working in Riyadh, Jeddah and Dharan (mainly for ARAMCO), and most are families;
They return back to the US for the summer holidays about now, and at this stage, most fly out with a BA service from either of those three cities, to LHR, where they then connect to a US-bound flight.
They'll all fly back in September. I'm sure TWA could do a roaring trade during the summer season, but as far as going to the US is concerned, that's it, until next summer.
At winter and easter, most foreign workers in Saudi, fly off to Dubai or South Africa/Australia, or drive across the causeway bridge to Bahrain for a week's holiday or more, but not back to their home countries.
That's where TWA would lose out-no traffic during the rest of the year-what are they going to do outside summer? Perhaps they should operate seasonally. Tourists are out of the question, there's no such thing in Saudi. Why don't they try out Dubai from Cairo or Tel Aviv? That would be more sensible economically.