When there is a periodontal gum infection, why doesn't a dentist just use a needle injection of antibiotics into gum than the long 7 day oral method?

Not that simple: Periodontal disease is a bacterial infection, but AB are not the first TX option. The periodontal infection is due to the presence of pathogens in the calculus that grows from bacterial plaque and the host immune response, the removal of submerged calculus and plaque in most cases is sufficient by itself to eliminate the secondary infection. Oral AB are given as an adjunct if needed.

Gum infection : Periodontitis a chronic disease. Acute infection in particular site is caused locally by multifactorial causes, usually by overgrowth of bacterial complex in the area. Debridement or though removal of bacteria and tartar in the area and assistance of antibiotic locally placed can help to manage the local infection. General periodontal maintenance and oral hygiene is needed to control disease.