Chanukah 5778

When one analyzes Chanukah along with its halachos and ideas, there are many comparisons from it to Succos. To name a few:

-The dispute between Beis Hillel and Beis Shammai in the Talmud (Shabbos 21b). Beis Hillel says we start with one candle and continue adding on each day of Chanukah while Beis Shammai says to start with eight and subtract a candle each day. Beis Hillel's reasoning is that we go up in sanctity and not down; Beis Shammai's reasoning is that it corresponds to the sacrifices given on Succos, which decreased in number each day.
-The Shulchan Aruch in Orach Chaim 671:6 notes that a Chanukiah must be placed lower than 20 amos (cubits); otherwise you don't fulfill the mitzvah of lighting Chanukah candles. The Mishna Berurah says this is because people won't notice it above that height. Similarly, The Shulchan Aruch in 633:1 rules that a Succah which is 20 amos tall is deemed invalid for use. The Gemara in Succah (2a) mentions three different reasons this is so.
-There is a strong emphasis on the concept of "hiddur", or beautifying a mitzvah with both holidays. On Chanukah, we follow the custom of doing the extra beautification of lighting candles, which reverts back to the dispute between Beis Hillel and Beis Shammai. On Succos, we look to use the most beautiful four species to fulfill our mitzvah of shaking them.

What's the connection between these two holidays? Why is there an overlap of similar ideas between these seemingly unrelated festivals?
The Aruch Hashulchan in Orach Chaim 670:5 gives a few reasons as to why Chanukah is eight days. His final reason is based on the verses from the Book of Maccabees, a scroll written from the time of the Second Temple and is part of the Apocrypha. He mentions that the Syrian Greeks had prohibited the Jews from bringing sacrifices during Succos and Shmini Atzeres. Once they defeated the Syrian Greeks, they restored the Temple and had decided to celebrate both Succos and Shmini Atzeres starting on the 25th of Kislev. They even brought lulavim. From then on, they decreed to keep those eight days as a holiday for generations (II Maccabees 10:5-8). So in essence, Chanukah is a make-up day for Succos.
My 12th grade rebbe, Rabbi Meir Soloveichik, noted an additional tidbit. In Al Hanissim, we mention how the Maccabees restored the Temple, purified everything, and "lit lamps in the courtyard." What was being lit? It couldn't have been the Menorah because that was in the inner chamber known as the "heichal"!
The Mishna in Succah (5:2-3) describes what would be done to prepare the Beis Hamikdash for the Simchas Beis Hashoevah, the water libation celebration. Once Yom Tov was over, they would place four golden candelabras in the courtyard and light them using old pieces of priests' clothing. These would illuminate all of Jerusalem. This must be what we are referring to in Al Hanissim. The Maccabees made sure to fully reenact the Succos festival they had missed. The Simchas Beis Hashoevah was the grandest celebration one could ever witness; how fitting it was to have one after defeating the Syrian Greeks and saving Judaism.