It would have taken longer to get there, but we would have still gotten there. As has already been mentioned, adaptation and speciation are already observed phenomena, and selection has long been a known mechanism for change. Genetics not only provide the explanation for how traits are passed, but also show the proof of common descent (for instance, through endogenous retroviral genes).
You’re right that we would lack a timeline, and it woould also make pathways much harder to trace, but the basics of heredity, adaptation, speciation and common descent are discernable without fossils. Fossils are very helpful, and were big, honking clues to set us on the right path, but are not absolutely essential.
I think you misunderstand the question. The claim isn’t that any human fossil will fit into a small box. The claim is that if you put every human fossil ever found, all together in one place, they would fit in a small box.
I think I know where this claim comes from. I recall reading something similar in a popular science book from the same era. Only there was a key qualification that either you or your source left out.
The qualification was that all the fossils between 4 and 10 million years ago could fit into a small box.
I don’t remember exactly where I read this. It may have been in Johanson’s book about Lucy or some other book about paleoanthropology from that era. Anyway, the main point the author was making was that for the key era where chimps and humans diverged, we have virtually no fossil evidence to point exactly when it happened. There was an argument going on at the time about this: those who looked at DNA thought it was more recent (4-5 million years ago) and those who looked at other data thought it was earlier (7-8 million years ago).
Note that all numbers above are from memory which has a history of being less than perfect.