Then why is he able to inherit the Hero's Clothes in the beginning of TP, and why can he use the Master Sword?
From Green Clothes, Hero's Clothes section:
"Faron notes that the Legendary Hero once wore these clothes, a treasure kept by the
Spirits of Light, and that it was only fitting that Link don the clothes as well, as his successor.
[6]" -Zelda Wiki
"
Look at your awakened form... The green tunic that is your garb once belonged to the ancient hero chosen by the gods... His power is yours. His is the true power that slept within you." — Faron (
Twilight Princess)
"
Only a person of the Knights Of Hyrule, who protected the royalty of Hylia, can become the Hero... You are of their blood-line, aren't you? Then you must rescue Zelda without fail." — A Maiden (
A Link to the Past)
From Hylia's Chosen Hero:
"As both of them are known to be the Goddess's chosen knights, most incarnations of Hylia's chosen hero will carry the bloodline of the
Knights of Hyrule who protects the
Royal Family of Hyrule,
[6][7] and therefore inherit the hero's spirit and courage.
[8] Future descendants of the chosen hero born into his knightly bloodline would be named after him for generations.
[9]" - Zelda Wiki.
Canonically, the only person allowed to use the Master Sword is an incarnation of Link. The Goddess Sword, which is what the Master Sword originally was, was created by Hylia so that her champion could use it, which just so happened to be the very original Link, AKA "Hylia's Chosen Hero.". The Master/Goddess Sword is
literally only for Link, and anyone that can use it is a Link. Twilight Princess Link can use the Master Sword, and is a reincarnation, so he counts as a part of the reincarnations.
Zelda (Hylia), Ganon (Demise), and Link (HCH) are always going to reincarnate:
"
Though this is not the end. My hate... never perishes. It is born anew in a cycle with no end! I will rise again. Those like you... Those who share the blood of the goddess and the spirit of the hero... They are eternally bound to this curse. An incarnation of my hatred shall ever follow your kind, dooming them to wander a blood-soaked sea of darkness for all time!" — Demise (
Skyward Sword)
Granted, this is said long after HCH is reincarnated into Skyward Sword Link, but regardless. Some Links are tied by blood, others not, but it doesn't change
they are all incarnations of HCH. So, in a way, you're right, but you're also wrong.