How critical is it to take antibiotics for a tooth abscess ?

Very helpful, but : The antibiotics alone will only mask the problem. You will need some form of dental treatment in combination with the antibiotics to eliminate the abcess.

Very Important: Infections will be gone when the source of the infection is gone, such as abscessed tooth, but antibiotics are very important to minimize infection. Local anesthesia cannot travel through infections so when you get "numbed" up, you'll still feel discomfort unless infection is gone.

Usually unnecessary: Proper and skilled root canal treatment of an infected tooth greatly decreases the need for antibiotics. In my opinion, it is always preferable to observe this happening because it allows us to differentiate that the therapy is working and not the short term benefit of an antibiotic. If there it advanced swelling, fever, or lymph node involvement, that is a different story.

Any infection: In my opinion, anyone with an infection, any place in the body should be given an appropriate course of antibiotics to both control the infection and prevent its spread. In addition to prescribing an antibiotic, the cause of the infection needs to be addressed with appropriate treatment.

Very important to control infection and prevent the spread of bacteria to other organ systems. Spreading bacteria can cause organ damage, even death. But antibiotics cannot cure, only control, infection until you run out of pills. Hands-on dental treatment is required.