This is a topic with very little in the way of records, at least for those in South Carolina. I doubt that I can help, but for what it's worth, here's what I've found while tracing my putative ancestor.
"Are there other Loyalist records where I might perhaps find names of his parents or siblings?"- The only Loyalist records I've found while searching for my ancestor are militia muster and pay records. These exist primarily in Canada and London, and some have been abstracted by various authors for specific purposes. Two example are books by Murtie June Clark and Bobby G. Moss. These records, if they exist for your ancestor, may help you determine exactly when and where he was present in SC or NC. But the records typically contain nothing of a genealogical nature, and given the peculiar spelling variants of the time, are often ambiguous.
"Did New York Volunteers who helped in the 'Southern Campaign' frequently stay in the Carolinas?" Anything is possible, but given that the Revolution in SC was primarily a brutal civil war, it seems unlikely that SC would have looked more attractive than NY to a contemporary Tory.
"Might he have stayed and had children in South Carolina and then moved on to Kentucky?" If he did stay in SC due to some unknown advantage specific to his situation, then this is indeed possible. With the general restlessness of the times, and the gradual decline in SC agricultural output, migration west was a common occurrence.
"Were these folks somewhat persecuted after they lost the war and he was intentionally leaving very little clue after arriving in Kentucky as to his past?" Grudges were held and old scores were settled for a very long time, and so it is likely that keeping quiet seemed a prudent course for Loyalists. My putative ancestor disappears from view after the fall of Charleston, and I've no doubt that he was trying very hard to to be difficult to trace. Probably yours was similarly inclined.