As has been conjectured by researchers that pathogenic bacteria from chronic gum disease might find their way to the brain and influence or cause dementia, could chronic viruses from inflamed gums seed virus to cause sites of infection not only in brain, but in other organs and tissues, like the cardiovascular system, kidney, liver, pancreas, thyroid, etc?

You are making a logical and accurate extrapolation here, to reason that if a chronic virus can be taken up from a deep pocket within the gums harboring an infection with virus, and get to the brain, it might well travel to other organs as well. There is always a possibility of lymphatic transport to spread far beyond an original nidus of infection, but uptake through breaches in the microvasculature are commonplace with periodontal disease, and those who suffer from this condition know all too well how frequently their gums will bleed because of disruption in the tissues and fragility with flossing or brushing the teeth. So viruses can spread bodywide, and what will determine the consequences will be a variety of local factors in various blood vessels and tissues enabling the virus to take up residence where it can trigger further propagation of new virus particles and be in a local environmental niche that is a stable, safe harbor for their growth and persistent presence.