It's a very familiar trope that a hero has to lose his personal history before he can start his quest. The absent parents mean TC has nothing tying him to the life of his everyday world.
TC doesn't lose his personal history, however. In fact, his relations to the two characters we know of in his life prior to contracting leprosy (Joan and Roger) break off from him, but their memories continue to torment him and Joan contacts him in TIW and reappears in the second chronicles. In addition, his relationships with people like his lawyer and the county sheriff impact him, both resolving and piling up problems for him in his world and, in the sheriff's case, magnifying his sense of isolation. Additionally, TC is tied to his identity as a leper, created by the conditions of the disease itself, his self loathing, and the reaction to the disease of other humans in his world
even after being cured of the disease in The Land.
Generally, even when he is in the land in the first chronicles, he is persistently marked by an attachment to the real world:
-In LFB, he has no particularly urgent business to get him back to earth, and early on sees his time in the land as illusory, but possibly healing on an emotional level nonetheless. However, when Lena poses the possibility that his world, which seems so horrible, is just an illusion, Covenant is horrified and even angry and sees Lena as an enemy engaged in a plot against his sanity (this reaction seems to be a prime motivating force behind Covenant's rape of Lena). After this point, he develops a strong dedication of refusing the reality of the land so that he does not become attached to it, and can continue to survive upon his assumed return to his world.
-In TIW, Joan calls Covenant shortly before he is summoned to the Land, and he angrily demands to be sent back by the Lords. However, they cannot do so.
-In TPtP, Covenant asks to be sent back by Mhoram so he can save the life of one girl, evne though the land's survival is presented to him as being at stake. Mhoram does so. After saving her, he returns to the land. Even at the end of the book, when the creator offers TC a life in the land as a hero, TC declines and asks to be sent back to Earth.
In short, characterizing TC as having 'nothing to tie him to the real world' simply because of a lack of parents seems to be a fundamental misunderstanding of him as a character.
(BTW, i have no idea about whether the second chronicles were in SRD's mind when he was writing the first, though it seems likely considering that they followed so closely afterwards).
I do suspect that maybe from the start SD had TC as a quasi-saviour, a quasi-redeemer, and for that reason also as a quasi-son of god.
In books, there are plenty of characters who are saviors and redeemers who aren't sons of god. Within the chronicles, Mhoram seems to be pretty solidly of human stock, but is probably more frequently a savior than Covenant (if for "smaller" stakes) through the first chronicles and is definitely more redemptive of those besides himself (in that Covenant actually could be said to have a corrupting influence on many characters in the first chronicles).
In a point that I already made that you haven't addressed, I observed that the creator seems to decide on Covenant during the opening of Lord Foul's Bane, as a result of Covenant's leper ideology/philosophy (for lack of a better term), specifically his stubborn refusal to give in to death. It seems highly unlikely that the creator would have a hand in Covenant's creation or birth, give him leprosy to make him an ideal champion, and then pull whatever strings were necessary to get him into the land; and it seems absurd that he would have a hand in his creation, forget about him, then find out decades later that Covenant was a suitable champion due to an unexpected diagnosis of leprosy and mark him to be summoned. If it was Foul, rather than the creator, that had a hand in Covenant's creation or life, it seems somewhat odd that he would have ended up being the perfect champion in the creator's eyes. Moreover, Foul's dialogue in Lord Foul's Bane seems to suggest an awareness that TC is the creator's champion, but other than that a relative unfamiliarity with Covenant.
I'm not going to address the spoilers on the last chrons since I haven't read the third book yet.