So when the infection is taken out of the gums by scaling, would taking amoxicillin help the process?

Scaling alone is considered by periodondists to prevent progression of gum disease.by removing plaque,but if gum infection is also present, antibiotics, by oral medications or by directly injecting them into the gums,.are probably going to be prescribed.

The infection is not completely gone with scaling. The antibiotic will help eliminate most of the bacteria but your home care if done daily will keep the infection under control and from spreading.

Not simple scaling. Root planing and curettage may be required. Specific antibiotics orally or by direct application may be advisable. The most important thing is altering your oral hygiene program as yours has not been effective. See a Periodontist. You can't do this with pills or computer clicks, See a gum/bone specialist.