What is Middle Age?
It is mostly about women, not men, since it is the reproductive span of women - in the context of lifespan, combined with the maturaltion rate of chidlren, that defines the basic cycles and stages of human life.
So, three reasons why Middle Age seems to have gotten later...
1. As a stage of life
If middle age is defined as being around the middle of life, perhaps typically the beginning of the second half of life; then when natural life expectancy is 70 years, 35 is half-way.
In modern times life expectancy is more like 80 years, with 40 as half-way.
If this second half is divided into middle and old age - then middle age runs from either about 35-53, or 40-60 - when old age commences.
2. The dwindling and ending of fertility
The average menopause is women is about 50, but fertility declines with increasing rapidity from 35 - and perhaps 40 is the limit for conception for most women.
So middle age could be defined as starting at about 40. The span of conception has been extended by advances in medicine, and by prenatal testing with selective abortion for abnormal fetuses - therefore, although female reproductive span has not changed much or at all, in practice the age of average and last conception is much older... delaying the onset of middle age.
3. The age at which offspring are mature and (culturally) ready to reproduce. This might be regarded as related to the average age of parents, especially the mother, at their first child.
In the past, the first child was usually about 20 (or less) but is now 30 (still increasing) - so middle age could be regarded as starting either as about 20 + 20 = 40 years old in the past, and 30 + 30 or 60 years old nowadays.
In other words, middle age is the age when you start having grandchildren.
Sixty years old seems absurdly late to be regarded as the onset of middle age - but I think this is more-or-less how it is now regarded; and to call even women in their fifties 'middle aged' would be regarded as an insult! - and I think this may be the reason why: Nowadays, women in their fifties frequently have children who have not (yet) married and not (yet) reproduced.
Until, or if, they have grandchildren, and themselves become grandmothers, women do not really feel themselves to be middle aged - hence the extraordinary delay in assigning the status of middle aged to women in modern society.