That cancer cell lines are often immortal is interesting circumstantial evidence for this.
Telomere shortening for example is hypothesized as one stop mechanism for runaway cell division.
My bet would be that this is part of the answer but not the whole answer. Biological systems rarely have one part match one function. Biology is not “designed” the way humans design things.
I read a paper about the depth of the tree of division of stem cells that also backs up the aging-is-side-effect-of-fighting-cancer. Sorry, I don't remember paper title or better keywords.