This is a good summary. Only thing to keep in mind is that an increase in genetic diversity need not imply a unidirectional march towards cancer, but an increase in risk. One of the most interesting paradoxes of cancer initiation research currently is the presence of "cancer-causing" mutations in phenotypically normal cells for decades prior to the appearance of the first cancer cell (see https://pubmed.ncbi.nlm.nih.gov/27059373/ for instance).
What's remarkable to me about this aging study and others like it is that they are able to reverse some aspect of aging despite the accumulation of genetic diversity with age as you point out. Perhaps what they're reversing was never really dependent on mutation accumulation with age, or that the presence of mutations does not fully explain the age-related degeneration they're interested in reversing.
as far as your genes are concerned, it's just as good if two of your children carry them on as if you carry them on yourself; there is no particular reason a priori for evolution to favor one over the other
however, your gametes form a genetic bottleneck which reduces genetic diversity (chimerism) among the cells of your children compared to the genetic diversity among your own cells
due to mutations during your lifespan, genetic diversity within your body increases, which increases the prevalence of genes that cause individual cells and tissues to compete with one another rather than cooperating; in the extreme, we call this 'cancer'
so we have genes like https://en.wikipedia.org/wiki/P53 which both cause aging (by suppressing stem cells, so you can't regenerate) and suppress cancer; more generally, many aging processes are known to suppress cancer directly
but more generally you can see genes' investment of resources in your offspring, rather than in you, as a way to protect that investment from cancer: if you get cancer, it almost never kills your offspring, so your genes can toss your worthless body onto the trash heap and continue on their merry way unmolested
of course you may feel differently about the importance of your own body surviving than (the optimization goals of the natural selection process selecting) your genes do