Difference between "season", "time of year", "time of the year"

Solution 1:

It is not appropriate to say, "Spring, Summer, Fall (Autumn) and Winter are times of (the) year." unless the noun time is specified (see below examples) as the noun season is marked by:

particular weather patterns and daylight hours. Time is not marked by them.

For example, there are only two seasons in Singapore, one is the dry season and the other is the rainy season. You can't replace season with time. It is not idiomatic.

Season could also mean:

A fixed time in the year when a particular sporting activity is pursued: 'the English cricket season is almost upon us.'

[Oxford Online Dictionary]

If you contrast hunting season with hunting time, the former is more idiomatic.

Regarding the noun time, you could use it in a sentence such as "Fall is the time (of year) for hunting". "Spring is the time (of year) when hunting is allowed".

If you use the definite article the before year, it could mean a specific year as in "The time of the year (2015) for hunting was Summer". The definite article is used to specify the year. If it is not used, it means non-specific year and could mean every year.

It's that time of the year for Iowa to get weird.

[Source: Rachel Maddow Show on MSNBC]

In the above sentence, that time of the year comes every four years when the first caucuses are held in Iowa. The year in the sentence is specified, e.g., 2012, 2016, 2020, 2024... and that's why the definite article the was used.

Note: The rules governing the definite articles are complicated. It is not easy to explain it briefly and the above explanation could never cover all the usages. It should be learned on a case-by-case basis.