I wrote this concept on the Avodah List many years ago. At that time a Chaveir pointed out that the Midrash Rabbah on Kohellet covers about 50% of this - Baruch shekivanti.
Q1. Why do we read Kohellet on Sukkot?
Q2. Why did Sh'lomoh HaMelech refer to himself as "Kohelet"?
1. Sukkot is a time of remembering Sukkat David, I.E. the Mishkan and the Mikdash
2. Sukkot is THE time for Hakheill
3 Sh'lomoh was THE first King of the Bayit Rishon
4. Ergo Sh'lomoh was [almost certainly] the FIRST to conduct Mitzvat Hakheil in the BH"M, and likely conducted the first "Hakheil" ever in history.
5. Thereby Sh'lomoh was THE "Kohelet" [the "Hakheiller"]
This means not so much preacher, but rather the gatherer of the People at the Mikdash - reminiscent of Moshe Rabbeinu gathering the people at Vayakheil and other occasions.
6 . Ergo we recall that first Hakheill by reading Kohellet on its anniversary, namely on Sukkot
7. We are thereby commemorating our first Mikdash, our First Hakheil in anticipation of the restoration of Our BH"M and of a NEW, Wise, Melech Hamoshiach [also a ben David] who exhibits the traits of [L'havdil] a "Philosopher King" - as well as the restoration of an era of Peace and Prosperity.
For Some Sources Supporting this Thesis Please See:
• Rashi on D'varim Vayelech 31:11 - where he presumes Melech, Beemah, and Azarah
• Kohellet Rabbah - Hamvo'ar edition esp. pp. 1, 5-6
The commentary there says
A. This corresponds to mitzvat Hakheil
And re: Kohelet/Sh'lomoh
B. "hu hamlech harishon asher kiyeim mitzvat hakheil zo b'veit hamikdash"