Does Korean need ~고 있다 and ~어 있다?

Korean verbs inherently have a continuous aspect about them, so the ~고 있다 and ~어 있다 patterns do not seem to be needed. Consider the following examples:

• 아이가 울고 있다 = 아이가 운다
  The baby is crying.
• 학생들이 공부하고 있다 = 학생들이 공부한다
  The students are studying.
• 비가 오고 있다 = 비가 온다
  It is raining.
• 내 아들은 학교에 가있다 = 내 아들은 학교에 갔다.
  My son is at school.
• 교실에 난로를 설치해 있다 = 교실에 난로을 설치했다.
  A heater is installed in the classroom.

As you can see from the above examples, the ~고 있다 and ~어 있다 patterns seem to be unnecessary.

Another interesting thing about the Korean language is that adverbs play an important role. For example, a simple adverb can change a sentence from present continuous to simple present tense. Consider the following examples:

• 아이가 운다.......................The baby is crying.
  아이가 자주 운다................The baby frequently cries.
• 학생들이 공부한다..............The students are studying.
  학생들이 매일 공부한다.......The students study daily.
• 비가 온다...........................It is raining.
  비가 자주 온다....................It frequently rains.

Can anyone think of a situation where ~고 있다 or ~어 있다 is necessary? Couldn't 산다 even replace a phrase like 살아 있다 (to be alive)?