“The integral age” view seems to be linked to Seder Olam – the Rabbinic View of Biblical Chronology re Moses; also to Rosh HaShanna – from the Gemara re the patriarchs; to Sefaria.org on Rosh HaShana and Kiddushin.
A Catholic article [extracts below] simply challenges the Rabbinic view and the Babylonian Talmud quotes regarding (specifically) the date of Moses’ birth and death being on the same day. Clearly, the origin of “the integral age” pre-dates the Catholic church by at least 2,000 years
From the National Catholic Register Update: The origin of “integral age” - Jon Sorensen noted that the phrase “integral age” may have been coined by William J. Tighe in this article. Tighe writes:
This belief seems to have been widespread in Judaism at the time of
Christ, but which, as it is nowhere taught in the Bible, has
completely fallen from the awareness of Christians. The idea is that
of the “integral age” of the great Jewish prophets: the idea that the
prophets of Israel died on the same dates as their birth or
Several statements in the Babylonian Talmud have been used to suggest
that Moses died on his birthday. This appears to be stated in at
least three places (b. Rosh Hashanah 1 [1:1, VIII.3.X], b. Sotah 12b
[1:8, III.38.Q], b. Kiddushin 39a [1:9, II:9:B])
Rosh Hashanah may apply this reasoning to Abraham, however this is unclear. In more recent times, it has been applied to David and perhaps other figures. However, the only person that the Talmud clearly applies this reasoning to is Moses.
Further, while the Talmud dates the claim that Moses was born and died on the seventh of Adar to the period between A.D. 10 and 220 (b. Kiddushin 1:9, II.9.A-B), the argument involving those who obey God living in whole year units may date to a few centuries later.
It can safely be claimed that some Jewish sages from approximately this period in history had the idea that some holy men (at least Moses) lived in whole year units and this may or may not have played a role in the thinking of early Christians in fixing certain feast days.
(Author Jimmy Akin is the Senior Apologist at Catholic Answers, a contributing editor to Catholic Answers Magazine, and a weekly guest on “Catholic Answers Live.”)
That's as much as I've been able to find out myself, but (ideally) those who hold to this view could give a much better answer, and I will leave this open in the hope that such an answer will eventually be posted.