An old person
"Old" is mostly how you view it. An older person is simply one who has lived a long time.
Like an old dog unable to learn new tricks
Perhaps you can't teach a dog new tricks, but let's talk about humans instead. Older persons can be educated and can enjoy learning new things.
Past their prime
Plenty of older adults contribute to society in a meaningful way as they grow older. Labeling a group as "past their prime" ignores individual differences that exist among people.
A victim of illness
Senility is an illness. All old do not experience senility because senility is a disease process that affects a portion of the old.
Unable to live long if their parents had not
There does seem to be some relationship between heredity and longevity, but heredity may have nothing to do with parents or grandparents dying at a relatively young age. Communicable diseases, accidents, suicide, and preventable diseases may have prematurely taken the life of a family member, but it does not mean that you will lack longevity in your life. Establishing good health habits can contribute to a long, healthy life.
If good health continues to old age, thus there are no disease processes in existence that affect mental functioning, the old have the advantage of adding their experiences into situations which can improve overall intelligence.
Lacking thought of financial security because they have Social Security to calm any fears
It was never the intent of Social Security to comprise all of an older person's income, thus the monthly benefit is not substantial enough to support all elders without the presence of other sources of income whether they be in the form of welfare, personal savings, or pension plans.
Age can bring wisdom, but it doesn't guarantee it. To become wise in old age one must work towards it, which isn't something that all people are willing to work towards.