The whole point is that ancient peoples, and especially Romans, are not a homogeneous group of people. Whether they define themselves as white is not what you should be arguing about, because most definitions of "white people" includes a diverse set of ethnicities as well.
WRT the Lybian Roman emperor: no one claims that ALL Roman citizen were white, or Italic, or European. Any such large Empire was heterogenous. But it is safe to say that a typical Roman citizen, espicially a Senator, would have been white for any reasonable definition of "whiteness", especially in a TV series. There was one black US president (his mother was white and father black if I remember correctly), but a typical US president so far was white.
I don't have anything against hiring a bunch of Chinese or Zimbabwean actors to play Senators, but that would certainly look odd. In the same way as hiring a bunch of Ukrainians to play Zulu warriors would look odd.
You also seem to be really caught up on senators. If you read the entire article, you'll note very little attention was given to senators.
Even so, that is not what the original argument was. The argument was that Rome was not homogeneous. Even in a heterogeneous society, there will be trends towards looking one way or another. The problem is simply with depicting all of Rome as (essentially) British.
Nobody has mentioned China or Zimbabwe, so this last argument is a lazy straw man.
We're circling back to the definition of "white". How's that a bold statement? Today's Great Britain has very diverse population, too, including lots of people from Africa, and yet I wouldn't think that a statement that a typical Brit is white is controversial. Typical Brit is not black, although there are many black British people, of course. And even if we constrain discussion to the most narrow definition of "white" (say, 18th century Englishmen) let's not forget that English people are the result of centuries of mixing of Britons, Germans, Celts... So, white Britons are themselves of diverse ancestry, but all these diverse ancestral people are "white". Let me repeat that race is an artificial construct, but if I'm pressed to say if someone is "white" or "not quite white" I can only go by outside appearance. Someone in Europe can have some genes from one Zimbabwean ancestor, but if he "looks white" to me, I can only say he's "white".
> The problem is simply with depicting all of Rome as (essentially) British.
Yes. That is a TV series. The point of TV series (in my opinion) is not to be 100% accurate, it's to entertain and (somewhat) educate. And the people in TV series are not real ancient Romans, but current actors. The series is for British audience, so it's normal that it's filled with British actors. If the series was produced in Bulgaria, the senators would be played by Bulgarian actors. Both are not exactly of the same color spectrum as ancient Romans, but are close enough to pass as Romans to their viewers. If the series was produced in China, and Romans were played by Chinese actors, that would probably pass as normal to Chinese audience. Maybe Zimbabwean actors could pass as Romans, at least in sub-Saharan Africa; why not? I'm not sure why people insist so much that British actors are bad representation of Romans. They seem quite OK to me (I'm not British).