From Merriam Webster: geriatric, a branch of medicine that deals with the problems and diseases of old age...
Not everyone old is geriatric:
I have a friend who was born in 1936, and five years ago we walked for an hour through field and forest in
Thailand, after which he proclaimed being tired. Body and mind in better shape than some people I know who are a generation younger than he. Old, but not geriatric.
Another friend was born in 1943, now uses crutches outside his room. We were on the way to a restaurant 500 m away, but due to fatigue we cut the trip short to a restaurant only 200 m away. Old and geriatric.
From Merriam Webster: venerable, calling forth respect through age, character, and attainments.
You can be venerable without being old, and you can be old without being venerable.
I can't find anything offensive with "elderly".