Gee, that's in the ball park but...
Leaving aside the question of personifying genes, which in truth don't
give a damn about anything, I do think you are coloring them a bit
harshly. There is some advantage to the gene in keeping mom and pop
alive, especially as long as they can breed with fidelity, it just
isn't a strong enough advantage to change the genes quickly. Most of
human evolution took place when old age was not a common cause of death
so it really did not help a lot to have a long natural life span. It's
not so much that genes don't want old folks around as that it's just
not high on their priority list.
To pursue this question seriously I would suggest Michael Rose's work
selectively breading fruit flies for longevity, Caleb Finch's
_Longevity, Senescence and the Genome_, and a good book on the
mathematics of evolution. I have spent a little time with each of
these but not enough to really answer Joao's question.