I assume the argument here is that if you see a huge linguistic expansion and a huge demographic expansion occurring at the same time, you can reasonably conclude that they are the same phenomenon and the expanding language is spoken by the expanding people. But the language and the people are not in general the same; people may voluntarily adopt a foreign language without moving (compare India and English) and they may fail to bring their language with them when they move. They cannot avoid bringing their DNA with them, but it will die out when they do, which their language may not.
That's a lot to assume, though. One possible counterexample is contemporary North America. The English-speaking population spread across the continent in a linked process, the same phenomenon, but the majority contribution genetically, is not from England. There are communities in North America where almost no one is of English ancestry, yet they are English-speaking.
The same sort of social/demographic upheavals and changes that can create large language spread, are also the same kind that can create homogenization towards a single language despite people speaking many languages.
> The Chinese of Thailand often cannot speak any Chinese (they speak Thai)
That was because of forced assimilation in the 1930s-50s by their proto-Fascist dictator.
> Yiddish is a Germanic language spoken by a non-Germanic people
Because there was plenty of intermarriage and the community was in Central Europe since Roman times
Amharic isn't the only Semitic language spoken in Ethiopia. We have other Semitic languages spoken both in Ethiopia and Eritrea. It is just the Red Sea that divides these two countries from the other Semitic speaking countries. Yemen, across the sea has other Semitic languages other than Arabic. Also, just like Indo-European is a large grouping of languages, the Semitic languages are ordinarily grouped under Afro-Asiatic languages. Thus, besides some of the Semitic languages spoken in Ethiopia and Eritrea, we also have two other subgroups of Afro-Asiatic namely, Cushitic and Omotic languages that are also spoken in Ethiopia. Amharic thus is well contextualised for Ethiopia.
Edit: also, if you look at pictures of Ethiopians and Eritreans and of Yemenites you’ll see they’re way more similar in appearance than Ethiopians and, say, Nigerians.
In the horn of Africa they've always spoken Semitic languages as far as we know, and actually it's possible the entire language family originated in Africa, not in West Asia as I guess you assume. This hasn't been proven one way or another, both theories exist.
The link is very strong for the obvious reason that everyone learns to speak to their parents.
My point is that things happen to break the link, not that the link can break for no reason.
And going back further there is evidence of genetic continuity with pre-Indo-European European populations too.
Prior to that it’s more likely that the invaders killed and/or subjugated most of the mean and put themselves at the top of local hierarchy meaning that they and their descendants were more likely to have successful offspring.
Based on genetics that seems to have been the case in Europe.
By that period in history, you'd usually call it Coptic, not (the much older) Ancient Egyptian.
(And really, as it evolved so dramatically over 3000 years, ‘Ancient Egyptian’ can’t easily be called a single language to start with. I tend to think of Coptic as simply the latest stage of Ancient Egyptian.)
* originator of the Anatolian hypothesis
It should also be noted marginally that haplogroups are, for similar reasons, a recent focus in history.