If aging is caused by "DNA Damage" why is the process of aging generally similar between people?

When I hear the phrase "DNA Damage" I imagine completely random changes in the DNA of a person’s cells. If this is the case, then it seems like every person would age differently depending on what happens to their DNA. But that is not what we see. In general people age in very similar, predicable ways almost as though it was programmed into the system.

Is it just that most random DNA changes result in cell death and that most of aging is just having fewer healthy cells in any given tissue?