Following Alexanders demise, during the Diadoch and Roman time, the people (at least the upper class) certainly had a strong Greek influence. But anyway, todays Egyptians are to the vast majority descendants of the invading Arabs.
I don't buy into the theory that Alexander's Greek troops had a significant impact on the gene pool
I'm ambivalent on this. The Graeco-Baktrian Kingdom in the Punjab region, following Alexander, had an evidently greek culture. Artworks show European features. There had to be Greeks among them, at least at the top. Today, as you said, there are people with bright eyes, light complexion and fair hair. I won't totally rule out an influence by Greeks in the region.
However your position is being corrobated by the fact that the Mazanderani and Gilaki people (North-Iranians living at the south coast of the Caspian) who are natives and had little intercourse with other people, have lighter complexion, eyes and hair.
Prominent example of a Mazanderani is the Ali Larijani, the Iranian regime's nuke negotiator:
Thanks for that tidbit. I had no idea.
Actually the Arabs were and are a minority in all so-called Arab lands except in Arabia. The Egyptians are native (Copts). They adopted the Arab language and Islam over centuries. Syria was still a Christian majority nation until the First Crusade. In fact the crusades backfired when the Latin west went on rampages against there Christian brethern in the near east. This drove lots of Christians to embrace Islam. It has been estimated that the whole Arab empire for the first hundred years was ruled by about 1% of the population (meaning Arab Muslims).
I personally think the US could exploit some of the latent nationalism in Iran and Egypt. Most Egyptians think of themselves as Egyptians first, Arabs second. The rest of the Arab world looks at them that way also. The US should be playing up the Persian aspect of the humilating loss to the Arabs. The Persians do not like the Arabs overall.