Egypt was under the control of the Arabs directly for six consecutive centuries, and then under the control of people who likely used Arabic for government functions due to the varied origins of the Mamluks themselves. So, in short, for almost a thousand years it was under the control of administrators who spoke Arabic, right in the middle of a large empire where Arabic was the main lingua franca. That is a huge time frame for the language to filter down to the general populace.

As to the second claim, the 'Arabization' of North Africa was mostly a cultural exchange rather than a replacement of peoples. Recent studies suggest that the average Egyptian can likely trace their heritage back thousands of years, and of course, clearly Berbers are a wholely distinct people from the Arabs. The Fertile crescent and Syria is a bit more, but there would have been Arabs and other Semitic peoples there before the Muslim invasions. The truth is that outside of a few isolated points in time, large scale population transfers just weren't that common. While the elites might be replaced, and the language of government and administration might change to reflect that, the change in the general population is generally going to reflect the cultural pressure from above rather than wholesale replacement of populations. This is not to say that some population transfers didn't take place in the Caliphate, but that they are almost certainly a minor component of the over-all Arabization of the Middle-East and North Africa.