I found a really excellent answer to this question in one of my university evolution textbooks [1]. The fact that men generally die sooner than woman (in humans) has been well studied and we've come to a fairly decent answer.

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Why do people die?

The answer to this mystery has been partially solved by senescence theory (Williams,
1957). Senescence is not a specific disease, but rather the deterioration of all bodily mechanisms as organisms grow older: Senescence theory starts with an interesting observation:
The power of natural selection decreases dramatically with increasing age. To understand
why this occurs, consider a twenty-year-old woman and a fifty-year-old woman. Selection
operates far more intensely on the younger woman, since anything that happens to her could
affect most of her future reproductive years. A gene activated at age twenty that weakened
a woman’s immune system, for example, could damage her entire reproductive capacity. If
the same damaging gene became activated in the fifty-year-old instead, it would have
almost no impact on the woman’s reproductive capacity. Selection operates only weakly on
the older woman, since most or all of her reproduction has already occurred (Nesse &
Williams, 1994).

Williams (1957) took this observation as a starting point and developed a pleiotropic
theory of senescence. Pleiotropy is the phenomenon whereby a gene can have two or more
different effects. Let’s say that there is a gene that boosts testosterone in men, causing them
to be more successful in competing with other men for status early in life, such as in their
twenties and thirties. But the elevated testosterone also has a negative effect later in life—
increasing the risk of prostate cancer. This pleiotropic gene can be favored by selection-—
that is, it increases in frequency in subsequent generations—because the early advantage in
status gains for men outweighs the later cost in lowered survival due to prostate cancer.
Through this pleiotropic process, we have evolved a number of genes that help us early in
life but cause damaging effects later in life, when selection is weak or absent.

The pleiotropic theory of senescence helps to explain not only why our organs all wear
out at roughly the same time late in life, but also why men die younger than women—
roughly seven years earlier on average (Kruger & Nesse, 2006; Williams & Nesse, 1991).
The effects of selection operate more strongly on men than on women because the reproductive variance of men is higher than that of women. Stated differently, most fertile women
reproduce, and the maximum number of children they can have is sharply restricted—
roughly twelve, for all practical purposes. Men, in contrast, can produce dozens of children
or be shut out of reproduction entirely. Because men have greater variability in reproduction, selection can operate more intensely on them than on women. In particular, selection
will favor genes that enable a man to compete successfully for mates early in life to be one
of the few who reproduces a lot or to avoid being excluded entirely.

Selection for men’s success in mate competition will be favored, even if it means
that these genes have detrimental effects on survival later in life. Even though men can and
sometimes do reproduce for a longer period of time than women. Senescence theory explains
why these later reproductive events will have a much smaller impact than events occurring
earlier in life for men. Genes will be selected for early success in mate competition more
strongly in men than in women, at the expense of genes that promote survival later. This
strong selection for early advantage produces a higher proportion of pleiotropic genes that
cause early death. As one researcher noted, “it seems likely that males suffer higher mortality
than do females because in the past they have enjoyed higher potential reproductive success,
and this has selected for traits that are positively associated with high reproductive success but at a cost of decreased survival” (Trivers. 1985, p. 314). Men, in short, are
“designed” to die sooner than women, and the theory of senescence helps to solve the
mystery of why.

In summary, selection is most potent early in life because any events that happen early
can affect the entire span of a person’s reproductive years. As people get older, however, the
power of selection weakens. In the extreme case something that happened to you in old age
right before you died would likely have no effect on your reproductive capacity. This means
that selection will favor adaptations that give beneficial effects early in life, even if they
come with heavy costs later on. These heavy costs cumulate in old age, resulting in the deterioration of all body parts at roughly the same time. In this sense organisms can be said to
be “designed” to die.

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The authors are pushing the X-linked recessive inheritance
hypothesis https://en.wikipedia.org/wiki/X-linked_recessive_inheritance , which by itself is fine but by itself does not paint the whole picture. The theory does not take into account how genes on the Y chromosome can affect the expression of other genes. For instance, a single gene on the Y chromosome can affect the expression of thousands of other genes across the whole genome.

[1] Evolutionary Psychology: The New Science of the Mind (4th Edition) by David Buss, pages 101-102