Why lab technicians use indirect (antibody reaction) method for diagnosing?
Short answer: Antibiotic treatment is important because the body can't always keep up with the infection even if it has launched a response.

Background:

First, let's look at what the antibody reaction method is actually doing.


Taking a sample of blood or body fluid
Testing the sample with known pathogens - viruses, bacteria, fungi, etc.
Identifying which infection causes a reaction.


This allows us to determine which infection we are dealing with. This can be especially useful when the infection is hidden in a place we are unable to easily swab - the stomach, heart, a small wound that has already healed. Sometimes this test is just used to monitor how well you are fighting the infection, but it also allows them to evaluate what the best course of treatment is for you. (Reference 1) 

Just because the body has an immune reaction doesn't mean that the body's immune system can actually finish the battle against a disease. In the example case of Strep A, you may be able to get over the infection yourself but it will take longer than if you are prescribed antibiotics against it. 

See this question on the CDC:


  Q: How common is invasive group A streptococcal disease?
  
  A: Approximately 9,000-11,500 cases of invasive GAS (Group A Streptococcus) disease occur each
  year in the United States, resulting in 1,000-1,800 deaths annually.



Note that invasive Strep A usually infects children, elderly, and  immunocompromised people (anyone with a weaker immune system typically). 


Unless you have absolutely no immune system at all, there will be an antibody response put out by your body. The NHS goes into detail about different parts of your immune system. 

Recall also that Strep A (our example) could divide faster than the body can keep up with. So we would need antibiotic help to fight the disease. Using the "indirect" method we can find a hidden infection. 

According to the NHS the antibiotics put a stop to growth and spread which allows the body to come in and finally finish them all off, or they disrupt a process the bacteria needs to survive and kill the infection for the body. Thereby helping the immune system (the body's best line of defence) finish off the infection when it couldn't do it on its own.