Why can't my child self-teach him/herself, i.e., working without a coach?

In fact, s/he can self-teach at various levels, some as a beginner, and some as an advanced player. Most self-teaching kids have extremely strong self-motivation to learning chess and strong work ethics. They don't like chess, instead they love chess. To them, chess is as important as lunch and dinner, maybe even more.
 
But most kids are not in this exetreme range. They themselves 

• can't know what needs to be done and at what time
  
• can't find appropriate material to study 
  
• can't setup high enough goals for themselves
  
• can't make their own plans 
  
• can't measure their own progress
  
• can't tell whether they are on a wrong path
  
• can't correct themselves quickly if they are on a wrong path
  
• can't figure out what is blocking their progress
  
• can't persist if their progress becomes stagnant
  

Therefore, they need some external help from a coach.

Even Bobby Fischer, who was famous for self-teaching himself chess and Russian (because he wanted to study Russian chess books), has a mentor, Jack Collins. All world champions and other top level tournament players hire seconds to help them train and practice. 

Everybody has blind spots s/he can't see by him/herself. For beginners, there are many, maybe too many.