In an alternative world where ancient Akkadian and Egyptian corpora didn't survive, if someone formulated the "Afroasiatic hypothesis" encompassing the branches that current consensus places in Afroasiatic, would they be able to prove their hypothesis? Specifically, would they be able to prove that Chadic languages should be included?
First of all, I should say, I don't think the presence or absence of Akkadian would make any significant difference. The Semitic family is widely attested and was well-understood even before Akkadian was deciphered—in fact, scholars' understanding of Semitic was the main reason why Akkadian could be deciphered at all! And while it's led to some new re-evaluation of how Proto-Semitic worked, it hasn't drastically changed our understanding of the family and its connections.
But to answer your specific question:
Specifically, would they be able to prove that Chadic languages should be included?
As best I can tell, the first person to propose a relationship between Chadic and other Afro-Asiatic languages was Leo Reinisch, in his 1909 Das persönliche Fürwort und die Verbalflexion in den Chamito-Semitischen Sprachen ("Personal Pronouns and Verbal Inflection in the Hamito-Semitic Languages"). He used some evidence from Egyptian in his work, but seems to have focused mostly on the Cushitic languages as a point of comparison, so it seems likely to me that he would have made this connection even without the Egyptian corpus.
Reinisch's work doesn't seem to have been especially popular, but M. Cohen also apparently compared Hausa (Chadic) specifically with Semitic, and was cited by Greenberg (Studies in African Linguistic Classification: IV Hamito-Semitic, 1950), who criticized his methodology but agreed with his conclusions:
Unfortunately, up to now, Hausa forms have been cited in isolation and without much consideration of the other languages in the area to which Hausa is clearly related. This is easy to understand inasmuch as Hausa is a conspicuous language, perhaps the most widely spoken in Negro Africa, and material on it is relatively abundant. The bringing in of other languages of the Lake Chad area greatly strengthens the case for the entire group of languages as forming a branch of Hamito-Semitic. Other languages in the area show pronominal and verbal forms similar to those of Hausa, thus showing that they are not recent borrowings from Arabic as has sometimes been suggested. Vocabulary resemblances are similarly strengthened and further etymologies become possible which could not be discovered on the basis of Hausa alone.
("Hamito-Semitic" being an obsolete grouping of mainly-African languages that was eventually reworked into Afro-Asiatic, and "Negro Africa" being an obsolete name for sub-Saharan Africa as a whole. Neither is an accepted term nowadays, but they were standard in Greenberg's era.)
So Egyptian doesn't seem to have been the key to connecting Chadic with the rest of Afro-Asiatic; I'm reasonably confident that the connection would have been made even without Egyptian evidence.