during the week(s) vs on the weekend(s)

• Our lives are divided into hours, days, weeks, months, and years.
• There are 52 weeks in a year.

Here the plural forms refer to facts that are indisputable and immutable.

In the NYT article, the journalist writes

During the week, Nate Brakeley works as a data analyst.

This is about a rugby player's routine, and the singular week specifically refers to the working week (BrEng) / workweek (AmEng) and there is strong evidence that suggests the singular form usually has the upperhand when it follows the preposition during.

During the morning / afternoon / night
During the day / week / month of May / year
During spring / summer / fall / winter

Merriam-webster defines during

1: throughout the duration of //swims every day during the summer

Cambridge's definition is

from the beginning to the end of a particular period. They work during the night and sleep by day.

Lexico offers these examples

• The mill, which was open to the public during the week, has had its visiting hours slashed.
• Walking through the city during the morning rush hour can be a bit of a battle.

The plural “during the springs” would not be ungrammatical but it tends to be highly unusual.

With the prepositions “on", "in" and "at" the plural form is more idiomatic.

In the mornings / afternoons / evenings

On Mondays, Tuesdays etc.

At the weekends (BrEng)

On the weekends (AmEng)
See Ngram chart

In the 1980s

With the exception of the seasons, which normally remains singular

In spring / summer / fall / winter

According to Ngram the phrases "in the weeks before" and "in the weeks leading to” are the most common but neither can replace "during the week” as the prepositions before and to are followed by a noun. By doing so the image the journalist seeks to evoke, one of the typical professional working in New York, is spoiled.

In the weeks before training, Nate Brakeley works as a data analyst…
In the weeks leading to the championship,…