"Luc Bovens, a philosopher at the London School of Economics, argues in the Journal of Medical Ethics that couples who try to prevent pregnancy by avoiding sex during the woman's most fertile time of month may be more likely to produce embryos that do not develop or implant in the womb. "

"Around this time of year for the last eight years, Mr. Connors has left his home in Silver City, N.M., and his wife, and moved into a two-room cabin on a mountaintop in the Gila National Forest "

"The 57th Street and Sixth Avenue station, for instance, is now served by one of four different trains, depending on the time of week."

"On eBay, the busiest time of day is from 6:30 to 10:30 in each time zone."

Why is "the" missing in "time of year", "time of month", "time of week", "time of day". I think "year", "month", "week", and "day" are COUNTABLE units of time. These examples seem to be missing an article after "of"......
The New York Times has its own style of writing and these are all examples of it. Newspapers are constantly searching for more space and eliminating the articles you have mentioned is one way of making room for additional words.
Is it proper grammar? You decide.
Newspapers get away with bad grammar?
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Since you access the NYT frequently, I'm guessing that you have a subscription to the paper. Once a month they publish a section on grammar where they admit to their mistakes and invite comments from readers. Yours is a good question to ask the editors.
I think NY Times is busy.
There must be something wrong.
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It seems to me perfectly normal, idiomatic English to say 'time of day',' time of year'.