Timothy doesn't try to make Phillip independent in only chapter 11. Timothy is constantly pushing Phillip to be able to do more and more stuff without Timothy's help. I think there are multiple reasons why Timothy does this, so it is up to you to figure out what you think is the main reason. I think the main reason is that Phillip understands the situation that they are in. The island isn't near main shipping lanes, and it doesn't have a plethora of food and water. Their chances for rescue are slim. Timothy also knows that his age is a factor to worry about. He is old. His body isn't as strong as it used to be. Injuries will take much longer to heal, and he is more susceptible to getting sick. Timothy knows that Phillip doesn't have a lot of survival know how; therefore, Timothy needs to teach Phillip a lot of survival stuff just in case Timothy dies or gets so injured that he can't help anymore. Phillip's blindness makes things more dire too. It would be one thing if Phillip was clueless and had all of his senses, but Phillip is blind too. Timothy wants Phillip to be independent in order to fend for himself if Timothy dies. Phillip also needs to learn some survival skills in case Timothy becomes incapacitated in some way. Finally, if Phillip is more independent, then Timothy's workload is lighter because both of them are pulling their own survival weight.

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.