The difference between the high prevalence of cancer in humans as a consequence of longer lifespan and menopause occurring due to the failure of the reproductive system does not at all seem clear.

I'll try to make it plain: high prevalence of cancer is the result of changes in the likelihood of cancer occurring during a person's lifetime, as a person ages their chances of getting cancer increase with each year. So mathematically, an older person is more likely to get cancer, but will not necessarily get it.
Menopause does not occur as a failure of the reproductive system, it is a natural part of the reproductive system. The chances of menopause occurring do not change linearly with age. Instead, a biological process governs its onset, and it will generally occur between the ages of 40 years and 60 years.

Are you saying that just because not every human gets cancer (versus literally every human female experiencing menopause), that's a significant difference?

Yes.
Again, cancer is an event based on a statistical probability, not a specific biological process that will occur in every human. Menopause is a biological event that occurs as a natural part of the female reproductive anatomy. To help you understand that, menopause does not occur when a female human runs out of eggs, it is not a failure of any kind, it is as natural as menarche.

If humans universally didn't live past 40-50 for the majority of our existence, and have only experienced longer lifespans for the past few hundred or thousand years, the argument for it being a consequence seems much more reliable (and a lot less of a 'just-so' story) than menopause being an adaptation.

Your premise here is flawed. You are confusing the measure of longevity with the measure life expectancy at birth. Life expectancy is an estimate of how much longer a person will live from a given age, so it changes with age. Death of young individuals that have not yet reached the age of reproduction was significantly greater in the past. This puts a significant skew on the measure called 'life expectancy at birth'. Because of that skew, it is not useful in making any good evolutionary argument about adult humans. Instead, you should look at life expectancy at adulthood for historically informative longevity.
There is no argument for menopause being a consequence of aging. It is not determined by a straightforward correlation with increasing age. It is a fixed process of biology.

To reiterate, it seems reasonable to assume that that percentage would increase if lifespan increased

No.
This is not reasonable. It is based a flawed premises and a poor understanding of biology.

The evolution of menopause is a long standing open question among evolutionary biologists. It poses a problem because 'fitness' describes only reproductive success. We would predict that reproductive success would be maximized by continuing to reproduce throughout the life-time of the adult female. But this is not the case. Instead, the adult female ceases reproductive effort on the creation of new offspring.