Timothy succeeds in making Philip independent because he is afraid that he is too reliant on him for survival. Timothy worries that he won't be around to take care of Philip. Plus, they bridge the gap between them with Philip coming to trust Timothy. This is an unusual experience for Philip who is both afraid of Timothy and prejudice of black people.
So Philip becoming independent with Timothy's help is very significant to the story.
He tried to make him independent for two distinct but related reasons. First, he tried to make Phillip independent as part of his general push to make him stronger. Timothy wanted the best for Phillip, and that included self-reliance. Second, Timothy knew he wasn't going to be around forever, and may have suspected he'd be gone soon. Therefore, for Phillip to survive, he had to be independent.