Since there are 28 actions grouped into 14 pairs presented in the format of a phrase עֵ֥ת...וְעֵ֣ת the question can be restated:
Question:
From the immediate context, what point is the author driving home by listing 14 pairs of opposing activities that seem to have their own merry-go-round of times and seasons?
Remember the Sabbath
For mankind, the most common and frequent reoccurring event is supposed to be the Sabbath:
“Remember the Sabbath day, to keep it holy. Six days you shall labor, and do all your work, but the seventh day is a Sabbath to the LORD your God. On it you shall not do any work, you, or your son, or your daughter, your male servant, or your female servant, or your livestock, or the sojourner who is within your gates. For in six days the LORD made heaven and earth, the sea, and all that is in them, and rested on the seventh day. Therefore the LORD blessed the Sabbath day and made it holy. (Exodus 20:8-10) [ESV]
Because the LORD God's work of creation lasted six days and He rested on the 7th day, man is remember the Sabbath every seventh day. The first time mankind would be able to observe this rest would be on the 14 th day. Therefore the number 14 is representative of the command to remember the Sabbath. Also, just as each day can be broken into two contrasting periods of Day and Night, each of the 14 pairs is made up of two contrasting items. This suggests the specific number of pairs in the passage is derived from the command to observe the Sabbath.
General Outline of the Passage
Each of the 28 actions can be considered as either positive or negative. This results in 14 pairs which are either positive/negative or negative/positive. When all of the pairs are mapped out, the overall arrangement generally follows an inverse parallel outline:
The primary division in this structure takes place after the 7th pair which means the 14 can be divided into two groups of 7. This general arrangement supports the idea the number of pairs was chosen based on the principles underlying the command to remember the Sabbath.
When the command to observe the Sabbath is applied retrospectively to the first man and woman, their day of rest occurred on the seventh day, immediately after their creation:
1 2 3 4 5 6 7 [1 2 3 4 5 6]
God: Work-Work-Work-Work-Work-Work-Rest
Man: Rest-Work-Work-Work-Work-Work-Work
Effectively, the first man and woman shared God's rest on the 7th day. This would begin an inverted Sabbath pattern (rest then work six days). The lack of a singular pair in the center of the structure in Ecclesiastes, results in a center which is shared by both the left and right side of the structure.
The symmetry of the structure changes at the 2nd and 3rd pair.1 On the opposite side there is a corresponding disruption in the symmetry at the 12th and 13th pairs:
Left side: 1 - [2 - 3] - 4 - 5 - 6 - 7
Right side: 1 - 2 - 3 - 4 - [5 - 6] - 7
Both "disruptions" correspond to apparent conflicts found in the second creation narrative:
These are the generations
of the heavens and the earth when they were created,
in the day that the LORD God made the earth and the heavens. (Genesis 2:4)
Then the LORD God said, “It is not good that the man should be alone; I will make him a helper fit for him.” Now out of the ground the LORD God had formed every beast of the field and every bird of the heavens and brought them to the man to see what he would call them. And whatever the man called every living creature, that was its name. (Genesis 2:18-19)
Where the first account describes the creation of the earth as spanning the second and third days, the second states this work was done on a single day.2 Where the first account states the animals were created before the man and woman; the second seems to say they were created after. This is particularly true of the birds which were created on the 5th day.
Thus the overall number and arrangement of the actions in Ecclesiastes support the idea the passage has been constructed following principles from the command to observe the Sabbath, which is derived from the work of creation described in Genesis 1 and 2.
Seasons
The introduction of the passage lacks any textual connection to either creation narrative:
For everything there is a season (זְמָ֑ן), and a time (וְעֵ֥ת) for every matter under heaven (3:1)
Neither זְמָן which means set or appointed time or עֵת which means time of an event or experience is found in the work of creation on the fourth day:
And God said, “Let there be lights in the expanse of the heavens to separate the day from the night. And let them be for signs and for seasons (וּלְמ֣וֹעֲדִ֔ים), and for days and years
(Genesis 1:14)
מוֹעֵד means appointed time, place, or meeting and Sabbaths are called as such:
The LORD spoke to Moses, saying, “Speak to the people of Israel and say to them, These are the appointed feasts (מוֹעֲדֵ֣י) of the LORD that you shall proclaim as holy convocations; they are my appointed feasts (מוֹעֲדָֽי). “Six days shall work be done, but on the seventh day is a Sabbath of solemn rest, a holy convocation. You shall do no work. It is a Sabbath to the LORD in all your dwelling places. (Leviticus 23:1-3)
Therefore while the passage in Ecclesiastes is based on God's work of creation as described in Genesis 1 and 2 and the command to remember the Sabbath, it is also presented as times and/or seasons which are not part of the creation narratives.
Conclusion
The reason Ecclesiastes has exactly 28 events which have been placed into pairs by עֵ֥ת...וְעֵ֣ת is that is the number and arrangement which corresponds to God's work of creation and the requirement for man to remember this work.
The main point of the structure contrasts man's work with God's. Unlike the LORD's Sabbaths which are scheduled and predictably determined by inanimate (created) objects, man's activities have "their own merry-go-around of times and season" as the OP observes.
Notes:
1. The assignment of some actions as positive or negative can be debated. For example war is negative, yet some wars could be considered as necessary and so positive (i.e. war against Hitler's Germany). The 2nd and 12th pairs are the two sets of actions which are the most subjective (in terms of being positive or negative). If 'plucking" means to harvest, it is a positive event; likewise silence and speaking can be either positive or negative depending on the context. Since this is true for both couplets, the symmetry of the structure is not affected. In fact the transition from positive/negative to negative/positive and vice versa is accomplished by couplets which are themselves "transitional" in that they contain actions which have the greatest subjectivity of being positive or negative.
2. If a day is 24-hours the conflict can be resolved by seeing the work as beginning on the second day and completed 24-hours later on the third day.
