An interesting idea, but probably not.

In general terms, the fate of Achilles tended to be understood in terms of the transience of glory and inevitability of death. When Odysseus sees the ghost of Achilles in the underworld, for example, he tries to comfort the great warrior's spirit by emphasizing the fame he had won in life. Achilles, however, replies that he would rather be the lowliest slave on earth than king of all the dead (Odyssey 11.486f) - glory, in other words, is no compensation for dying young.

The idea that Achilles' heel was the only vulnerable part of his body seems to have been a relatively late addition to his myth. Homer never mentions it; and not until the Roman period do we get explicit references to the heel as the source of Achilles' weakness and death (e.g. Statius, Achilleid 1.134). The tradition is probably older - some Greek vase paintings seem to show Achilles being wounded in the foot or heel - but for our purposes, the important thing is that no Greek or Roman author ever bothered to comment on the fact that the site of Achilles' injury corresponded to a tendon that real warriors were in danger of pulling.

Ancient doctors were well aware of the Achilles tendon, but they merely called it the "great tendon." The seventeenth-century Flemish anatomist Phillippe Verheyen seems to have been the first to describe it as the Achilles tendon.

​

Klenerman, L. "The Early History of tendo Achillis and its Rupture." The Journal of Bone and Joint Surgery 89 (2007): 545–547

The shortest answer is that we don't, and can't know what the origins of the myth are. The stories come from too far back in time, and are both too detailed (because they include things that would distract from a simple moral of "don't overexert yourself," and include the idea of his mother holding him by the heel as a reason for it to be vulnerable) and not specific enough (since the moral that you're proposing does not occur, and there is no direct evidence of damage to Achilles' hell making him lame or vulnerable, just that he's killed by it).

What you're proposing is called rationalizing a myth, which has a history going back to antiquity. Greek scholars of the Classical and Hellenic eras often looked back at their myths and try to explain away some of the more absurd or mystic aspects of the myths. Herodotus thought Paris may have kidnapped Helen because it was part of a pattern of abductions, like Europa or Io being taken from Greece to the lands east, while Medea was taken from the east back to the west. Sophocles said in the Phaedra that the myth of Orithyia being taken by Boreas is a romantic way of saying Orithyia was blown off a cliff by strong winds. Another similar rationalization would be the claim that the Jewish restriction on eating pork comes from fear of trichinosis or some other swine-borne disease, and that it was a health concern.

We do not know enough about the author of the story of Achilles getting shot in the ankle to determine what his intent was in writing the piece. (Even if you believe that Homer was a single person, the idea of an Achilles with invulnerable skin is not found in either of his poems.) We also don't know enough about his times to be able to determine what he knew about physiology, the prevalence of Achilles' tendon injuries, or their effect on the battlefield to know if that was a well-enough known concern to make it into a story.

Another thing that you have to consider is that Achilles was not the only figure in Greek myth who had a goddess treat his body in a way to make him invulnerable: Demeter, when she was out looking for her daughter Persephone, was engaged as a nurse to two children, Demophon and Triptolemus. Every night she would rub ambrosia on Demophon and put him in a fire to burn away the mortal parts of him. Before she finished, Demophon's mother saw Demeter put the baby in the fire and cried out, which made Demeter flee and left Demophon mortal. While the details are different, the outline of the story is similar. Isn't it more likely that it's the same story tied to a more famous figure? This happens more than you'd think in Greek myths, which most people don't realize because they only read a small selection of the existing corpus.

So the simple answer is that it is unlikely that the story of Achilles' heel is connected with a torn Achilles tendon, but it is impossible to know for sure.

References:
Homer, The Iliad
Homer, The Odyssey
Apollodorus, The Library

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