You are conflating two different dental problems, tooth decay and gum disease.
Sugar doesn't cause gum disease. Poor dental hygiene causes gum disease. Poor dental hygiene also causes tooth decay, but that doesn't make them the same thing. Tooth decay is caused by sugars, though it's mitigated by cleaning and fluoride. For tooth decay to infect the gums you'd have to be completely ignoring tooth hygiene. A dentist should have filled those cavities long, long before that point, or performed a root canal if you let it go that far.
Eliminating sugar won't change your chance of gum disease one way or the other. It's a bacterial infection that grows into plaques, and they have to be removed. People in the past developed gum disease long before sugars: Otzi the Iceman had gum disease.
The main modern cause of gum disease is smoking, though thankfully that is decreasing. Diabetes is also a cause of gum disease, and managing sugars reduces the chance of diabetes type 2. (Diabetes type 1 is an autoimmune disorder independent of your sugar consumption, though type 1 diabetics also have to manage their sugar consumption -- though certainly not eliminate it.)
The "sugar is evil" trope is just not helpful, and it's not applicable here. It does not cause gum disease and is not implicated in Alzheimer's by this study. People should consume less of it, but it's not the villain it's made out to be by Robert Lustig and company.