3rd Day of Chol HaMoed
The following words of the Ramchal will help to explain why it is davka Sukkot that the Christians have tried to grab hold of and make their own. But, whether they mean to bring us down to their level or raise themselves above their station, those same words will also explain what an impossible task they've set themselves.
Derech Hashem, Part IV, chapter 8, section 2:Yesterday's Christians usurped the Children's heritage via "replacement theology." Today's Christians have only just now agreed to move over and let the Jews sit beside them as an equal partner (conditional, of course, on a future acceptance of JC). The only way they will ever accept the truth that they have no claim on the Children's heritage at all is to give up Christianity altogether.
Sukkos observances in general are related to the Clouds of Glory.
Besides the physical benefit of providing shelter and protection, these Clouds caused Israel to be set apart and elevated physically, they likewise were responsible for the transmission of the Essence of Illumination that made them unique. As a result, they were differentiated from all peoples and literally elevated and removed from the physical world itself. Israel thus became the highest of all the nations of the world.
It was through these Clouds that Israel attained the high level that was meant for them. The result of this was then transmitted to every Jew for all generations. This is the Light of holiness, transmitted by God, which surrounds every righteous man of Israel, distinguishing him from all other individuals, and raising and elevating him above them all. This is the concept that is renewed every Sukkos through the sukkah itself.
The Light of God also shines over Israel and engulfs them in such a manner that when they take the lulav and its associated species their enemies are terrified of them. Regarding this, the Torah says (Devarim 28:19), "All the nations of the earth shall see that God's Name is called upon you and they will fear you."
This concept would have been openly attained immediately if it were not prevented by sin. Nevertheless, however, it still exists [potentially], and will be translated into action at the appropriate time.
This concept is completed through the various observances associated with the lulav, such as when we shake it and march around [the synagogue] with it. God's Kingdom over Israel is strengthened, ultimately causing their enemies to fall and be subjugated before them, until they finally choose to be their slaves of their own accord. This is the significance of such prophecies as (Yeshayahu 49:23), "With their faces to the earth, they will bow down to you..." and (Ibid. 60:14), "The sons of those who persecuted you shall come to you bowing." All the nations will then be subjugated and bow down to Israel, and through the Jews they will partake of God's Light that shines upon Israel. All their pride will be brought low, they will be humbled under Israel, and through the Jews, they will return to God. This is the general significance of the lulav and its details.
...The Talmud describes how, in the World to Come, G-d will challenge the gentiles to observe the commandment of Sukkah. Due to the intense heat, they will abandon the Sukkah, even kicking it on their way out. The Talmud notes that a Jew, too, is exempt from remaining in the Sukkah if he is uncomfortable, but he still does not treat the Sukkah demeaningly. Why is that difference so significant?
The Rem”a (Orach Chaim 639:7) writes, ‘Anyone who is exempt from sitting in the sukkah will not be especially rewarded if he remains in it.’ Accordingly, the “scorching hot Sukkah challenge” tests one’s response to losing the opportunity of being rewarded for a mitzvah. Therein lies the difference between the Jew and the gentile. Once the heat renders sitting in the Sukkah a “non-profitable” endeavor, the gentiles will kick the Sukkah, considering it worthless. The Jew, however, does not see the Sukkah as worthless even when he stands to receive no reward for sitting in it. Rather, he sees the Sukkah primarily as a means to fulfill G-d’s will: to sit in it when G-d so desires, or not to, if G-d’s law should deem it unnecessary. The reward is only secondary, so its loss does not change the Sukkah’s value.
—Reshimos 62, pp. 22-23 (Source)