It is true that there are many different types of cancer. I've often been the advocate for "you can't 'cure cancer' because it's not a single thing" , and on many levels it is not.
But I've come to change the conclusion that a single, overriding cure for most-or-all cancers is not feasable. At the root, the same basic event (uncontrolled division) is occuring. Given that there are even mammals which have developed nigh-immunity to all cancers (naked mole rats); it doesn't seem unreasonable that such a universal solution might be available to medical science at some point in the future.
As to the "accumulating mutations" problem. There's an unbroken line of cellular replication between the egg that formed you and the first DNA-based life to form in the oceans 4 billion years ago. Yes: there has been mutation in some of the clones of that cell that have terminated individual creatures over that time, making my scenerio a bit hyperbolic; but I do think that this "accumulation of mutation" within centuries is over-rated and my be (primarily) solvable as well.
If we removed non-cancer from a naked mole rat: how long before it got cancer?