As the title asks, why didn't the basilisk bite received in the Chamber of Secrets destroy the Horcrux inside the bitten person?
Yes, I know Fawkes quite quickly healed the bite wound with his phoenix tears, but when Harry used the basilisk fang to destroy Tom Riddle's diary the destruction of that Horcrux was immediate. When Ron hit the locket with the Sword of Gryffindor, it was destroyed immediately. Although the destruction of the ring occurred off-page, Dumbledore used the Sword of Gryffindor to destroy it, so presumably, like the locket, it was gone right away. Nagini died straightaway when Neville decapitated her. The destruction of the Hufflepuff cup happens off-page. The diadem seems to have been destroyed by Fiendfyre immediately as well, as when Harry examined it outside the Room of Requirement it was already leaking a blood-like substance, and then it cracked in half. When Voldemort hit Harry with Avada Kedavra in the forest, Harry woke up in limbo with the last fragment of Voldemort's soul - that awful flayed baby-like being - separated from him immediately.
Is there a logical/canon-based explanation for why the fragment of Voldemort's soul residing in Harry wasn't destroyed upon Harry being bitten by the basilisk?
NOTE: I've read the HP Wiki's entry on Horcruxes; however, I find the HP Wiki to be inconsistent in its accuracy, so I'm hesitant to accept the Wiki's explanation on the matter. I'd prefer an explanation constructed from information in the books or a statement by JKR, or the like.