The Rambam (Moreh 3:43) says exactly the opposite of the Tur: that this season was picked because it's neither too hot nor too cold for a Succah (in Eretz Yisrael's climate, of course).
These are unsatisfying answers. We are taught that our holidays are not mere tableaux vivants re-enactments, but instead a re-experience of the mazal and segula of the moment, a wellspring from which we draw spiritual strength. It is hard to accept the Rambam's and the Tur's assertions that Sukkos has no inherent temporal/spiritual relation to its place in the calendar.
Addressing this issue, the Vilner Gaon (born 1720) and the Chasam Sofer (born 1762) say identical answers. The Vilner Gaon is in Shir Hashirim 1:4, here. The Chasam Sofer is in his Drashos, page 29a, in the Drasha for Shabbos Shuva of תקצ"ד, D'H B'Yalkut, here.
Thank you, Heather H., for pointing out that today, October 17, is the Gaon's Yahrtzeit, 19 Tishrei Tof Kuf Nun Ches/1797. The Chasam Sofer's Yahrtzeit, 25 Tishrei Tof Reish/1839, is next Sunday, October 23. May they be meilitzei yosher for us and all Klal Yisrael.
They answer that the Clouds which (according to Rebbi Eliezer) the Sukkos represent, departed upon the sin of the Eigel. The Clouds only returned after we did teshuva and began building the Mishkan. When was that? Our teshuva was accepted on Yom Kippur, the tenth. On the eleventh, Moshe Rabbeinu gave us the commandment to build the Mishkan; on the twelfth and thirteenth, we collected the material and money we needed; on the fourteenth, the calculations, assignments and disbursements were made, and on the fifteenth of Tishrei, the work began. It is not just the Ananim we are celebrating- we're celebrating Hashem's return of the Ananei Hakavod as a sign of His love after we did teshuva. The Teshuva did not merely erase our sins, it reinstated Hashem's love. It is the great joy of the return of the ananei hakavod that we are celebrating.
The problem with this teretz is the pessukim in Nechemiah 9:19-20, which say
יח אף כי עשו להם עגל מסכה ויאמרו זה אלהיך אשר העלך ממצרים ויעשו נאצות גדלות. יט ואתה ברחמיך הרבים לא עזבתם במדבר את עמוד הענן לא סר מעליהם ביומם להנחתם בהדרך ואת-עמוד האש בלילה להאיר להם ואת-הדרך אשר ילכו בה
Even when they made the Eigel and said "This is your god that took you out of Egypt", and they did great blasphemies, You, in Your great mercy, did not leave them in the Desert. The Pillar of Cloud did not move away from them in the day to lead them on the path, nor the Pillar of Fire to illuminate the road they walked upon.
It appears that these pesukim, which clearly state that the sin of the Eigel did not cause the Cloud to go away, contradict the assertion of the Gaon and the Chasam Sofer.
The answer is, of course, that there is a difference between the Amud Ha'anan and the Ananei Hakavod. The Pillar of Cloud remained with them, but the Clouds of Glory, which covered and protected them, did disappear, and only came back when they began building the Mishkan.
You might disagree with this distinction. After all, the Aruch Hashulchan seems to commingle the two Ananim:
Also, we find that Chazal equate the Amud Ha'anan and the Ananei Hakavod, in Medrash Bamidbar Rabbah II:
Also, see (sorry about the transcription errors: my secretary stepped out for a moment...) Mechilta D'Rebbi Yishmael in Beshalach, 13:20, where it says very clearly
It's possible that some ananim went away and some didn't. But the pashtus of the Passuk in Nechemiah is that all of the Ananim remained after the sin of the Eigel, even before Yom Kippur. So what's pshat in the teretz of Gaon and the Chasam Sofer? We can find the answer to this question in the Chasam Sofer, here, (Drashos, page 53b, D'H Sha'alah).
The Chasam Sofer asks, why does the Gemara in Sukkos 26b not suggest that women ought to be Chayav in Sukka because they were involved in the miracle no less than the men? He answers, as he said earlier, that Sukkos commemorates not the Clouds, but the return of the Clouds after Teshuva. He says that the Clouds did not actually disappear, but they ejected any sinner that attempted to enter into them. The men, who were excommunicated for their sin of the Eigel, remained outside of the cloud until they began building the Mishkan. The women, on the other hand, had never participated in the Eigel, and were never driven away from the Anan. While any man entering the Anan were spit out, the women were able to enter the Anan at will. Since for the women nothing changed, the redemptive experience of Sukkos did not apply to them at all, and we certainly could not say they're chayav on the basis of אף הן היו באותו הנס. They weren't באותו הנס at all.
According to the Chasam Sofer, we can understand the pesukim in Nechemiah to mean that the Anan never went away, and still guided them. But it did not allow the men to enter into it, it was poleit them. Only when they began building the Mishkan were the men allowed to enter into the cloud, and it is this redemptive experience that took place on the fifteenth of Tishrei that we celebrate.
I want to point out an irony. According to the Chasam Sofer, the reason that we wouldn't apply אף הן היו באותו הנס to obligate women to do the Mitzva of Sukka is because they never sinned, they never were kicked out, and they never earned redemption. Sukkos is the holiday that commemorates a fall and a return to Hashem's love. They never fell, and Hashem loved them all the time, so they have no reconciliation to celebrate. Although it sounds strange, it makes perfect sense. The descendants of an alternative Avraham Avinu who never said "Bamah Eida" would never have been redeemed from Mitzrayim. I recently read that a physician lamented how he spends half an hour teaching a patient about lifestyle changes that will ensure that his heart remains healthy, and a cardiac surgeon that repairs a heart damaged by by a bad lifestyle is paid thirty to forty times as much for his half hour. Somehow, that reminded me of the Chasam Sofer's teretz.