Yes that is right.
It is true that common descent is the only natural way for organisms to emerge. But the data also fits what we expect if common descent is true. If it did not, the situation might be very different. It is hard to predict what science would do if there was clear and obvious evidence that humans arose by a non-common descent process. We can imagine what that might be…
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Perhaps human genomes could have been totally different from other life, clearly separable.
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Perhaps human biochemistry could have been totally discontinuous, as in our amino acids and nucleic acids were the mirror version of all life around us.
These are both examples of data that could have, in principle, definitively proven that humans and apes do not share common ancestors. Science, under its current rules, would like got to more exotic explanations, like aliens and such. As unlikely as that is, they might be forced their if the evidence against common descent were strong. Or maybe then their might be a rule change, specifically for this case. We are talking about a fictional world though, so it is hard to know.
Certainly common descent is falsifiable, at least in principle. It has not been falsified, because the data just matches up so closely. At the very least, we have to grant that it appears we share common ancestors with the great Apes, even if this is not ultimately the case.
Good questions @Ashwin_s