I agree with you as well, but the buddha also said in suttas
"What, bhikkhus, is aging-and-death? The aging of the various beings in the various orders of beings, their growing old, brokenness of teeth, greyness of hair, wrinkling of skin, decline of vitality, degeneration of the faculties: this is called aging. The passing away of the various beings from the various orders of beings, their perishing, their break up, disappearance, mortality, death, completion of time, the break up of the aggregates, the laying down of the carcass: this is called death. Thus this aging and this death are together called aging-and-death."
It is hard to read this in dhamma language terms and can be seen to only explain the usual physical ageing and dying. To my knowledge he says this to bhikkhus not to laypersons.
Unbowed, Unbent, Unbroken