The claim was that the bulk of the life expectancy number being driven by the mortality rate prior to adult hood was an "urban myth". Life expectancy in adult hood is a ~1.4x increase. Life expectancy from birth is a 4x increase. I'd say the bulk is driven by saving the lives of children and is not an urban myth.
Since the time period you chose, life expectancy at birth has increased by about 40 to 55 years. Near-elimination of infant mortality accounts for 15–30 years of increase, while increases in adult life expectancy account for an additional 25 years.
Can you really look those numbers and say that one of them is highly significant while the other "has not changed much"?
Translating this into percentages or "fold" increases is misleading because you are using a different denominator for the two increases. E.g. you are claiming that reduced infant mortality is a 2.2-fold increase (55/25) while reduced adult mortality is "only" a 1.45-fold increase (80/55), even though the number of years gained is almost the same.
And if you use the other end of the 25–40 year range, then reduced infant mortality is "only" a 1.38-fold increase (55/40).
Your second link shows years of additional life expectancy. For example, at age 80 it shows 8.43 years of life expectancy, meaning that the average 80-year-old lives to age 88.43.
So the average 25-year-old in the US has 52–57 years of life remaining, and will live to age 77 or 82, according to your link.
This is a very big increase over the pre-twentieth-century adult life expectancy of ~55 years of age! You claim that "life expectancy of adults hasn’t changed much" but your own links show that it increased about 50% in very recent history.