Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
Further linguistics suggest an origin in South/Central Africa but there is less physical evidence of this. Compelling linguistics and other cultural evidence suggests Ethiopia as the most recent origin point in Africa, and fossilized and pre-Human and early human finds are strong evidence of an Ethiopian origin.
It's important to make clear that the only migration linguistics debates is that of the Afro-Asiatic languages, which originated in northeast Africa 8,000 years ago or so, breaking up into the Semitic, Cushitic, Berber, etc. languages. No reputable linguist believes that there is any linguistic evidence remaining from as far back as when humans first left Africa many tens of thousands of years ago.
That is only partially accurate, though I am very much impressed at your understanding of the evolution of Semitic languages. Indeed, even just in the past 15 years, it has become the rather conclusive opinion that all Semitic language (and also culture) originated from a proto-Semitic language in Central Ethiopia, by the way which fits entirely into the traditional Ethiopian history which teaches that Ge'ez, the mother tongue of the Tewahedo Church, is in fact the mother tongue of all Semitic languages, and some scholars agree in part.
The clinching evidence is the immense diversity of Semitic languages even today in Ethiopia, in most Semitic places such as the Arabian peninsula and the Near East, there are around a dozen or so Semitic languages with around two dozen dialects, where as in Ethiopia there are nearly 30-40 individual, specifically Semitic languages and almost a hundred Semitic dialectic variations, which is clear evidence that the Semitic language has been evolving for a much longer period of time in Ethiopia than any other places where there is less diversity within the same language. Just as it is clear that Romance languages evolved in Europe because there are far more variations of these Indo-European languages.
But there is indeed linguistic evidence of early human history it the form of audible clicks and heavy glotallization. The glotallized sounds of Semitic languages (the "hawking" sound in Hebrew/Arabic) is the evolutionary leftovers from the very clear and prominent clicking sounds in the Khoi-San peoples are South/Central Africa, who DNA suggests are the oldest continuous living group of humans in the world. Indeed, almost EVERY single human being today contains the same exact DNA at the bottom of the chain as any living Khoi-San, which is striking evidence that their family is the beginning of all of our family tree. The combination of DNA and clicking languages strongly suggests the Khoi-San and other related populations as being our most common ancestors.
In Semitic Ethiopian languages there are indeed such heavy glottalizations that they are in fact clicks. I am learning Amharic, and there are 9 glottalized consonants, and it was very hard for me as a native-English speaker (which has not glottalizations) to learn to be able to do it, but in the end it becomes second nature, and in the hard K/Q sound the glottalization is so pronounced that it indeed clicks almost has obviously as the clicking languages of the Khoi-San of today.
The fact that within Ethiopia, already home to evidence of the oldest pre-human and human physical remains, and also the clear evidence of strong linguistic evolution and origin of Semitic tongues, and that these Ethiopian Semitic tongues even still retain a solid clicking sound, is probably the most clinching evidence of the Out of Africa theory.