What is the "Sukkat David"?

One of the "Harahamans" added for Sukkot (for Ashkenazim, the only Harahaman added for Sukkot) is:

הרחמן הוא יקים לנו את סוכת דוד הנופלת

We pray that The Merciful One will reestablish for us the Sukkah of David that is fallen.

This prayer is taken from the verse in Amos 9:11 (no connection to 9/11/01, but I can understand those who want to make such a connection):

בַּיּוֹם הַהוּא, אָקִים אֶת-סֻכַּת דָּוִיד הַנֹּפֶלֶת; וְגָדַרְתִּי אֶת-פִּרְצֵיהֶן, וַהֲרִסֹתָיו אָקִים, וּבְנִיתִיהָ, כִּימֵי עוֹלָם

In that day will I raise up the tabernacle of David that is fallen, and close up the breaches thereof, and I will raise up his ruins, and I will build it as in the days of old

Yeshivot Bnei Akiva has a nice article (in Hebrew) on the various interpretations of this verse.  Some say it refers to the Beit Hamikdash.  Others say that it refers to the Davidic Dynasty.  You can see some of these interpretations inside here.

The last opinion mentioned in the article is that of the Malbim.  The Malbim agrees that it's referring to the Davidic Dynasty, and adds that the Davidic Dynasty fell in two hops:

1. At first, we had the real Davidic Dynasty in its full glory - the descendants of King David were kings.
2. When the first Beit Hamikdash was destroyed, we no longer had kings from King David, but only Nesi'im that came from King David.  The kingdom was like a Sukkah - in a temporary phase.
3. In the past 1000 years, there is no authority to King David's descendants whatsoever as we don't even know who is from King David any longer.

The rebuilding of the kingdom will need to be done in three stages (the article uses the parts of the verse a bit differently, but I think this is what the Malbim means):

1. At first, Hashem will establish a Nasi for us to lead us in a temporary way like a Sukkah (אָקִים אֶת-סֻכַּת דָּוִיד הַנֹּפֶלֶת), but the Sukkah is still in ruins
2. Then, that leader will return people toward Hashem, thereby repairing the breaches and restoring what is broken (וְגָדַרְתִּי אֶת-פִּרְצֵיהֶן, וַהֲרִסֹתָיו אָקִים)
3. Finally, that leader will become a more permanent leader.  Namely, the King Mashiah.(וּבְנִיתִיהָ, כִּימֵי עוֹלָם)

The Malbim brings another verse to support this idea.  Yehezkel 34:23-24:

הוּא יִרְעֶה אֹתָם, וְהוּא-יִהְיֶה לָהֶן לְרֹעֶה...

 וַאֲנִי יְהוָה, אֶהְיֶה לָהֶם לֵאלֹהִים, וְעַבְדִּי דָוִד, נָשִׂיא בְתוֹכָם

First, he will be their leader.  Then, when he repairs the breaches and people start keeping the Torah, he will be accepted by the people as a leader.  Later, when the Kingdom of Heaven is revealed to all, the King Mashiah will reign as a permanent leader.

May both verses come to fruition speedily in our days.