R' Aharon Lichtenstein: mitzvos that we happen to do on a chag vs mitzvos that establish the character of the chag

 Y"T fast approaches, but wanted to get this in quickly.  The gemara (Suk 48a) says the Shmini Atzeret is רגל בפני עצמו לענין פז״ר קש״ב  Rashi on the spot comments רגל בפני עצמו  שאין יושבין בסוכה.  

Ramban raises a number of questions on Rashi, among them:

עוד שאינו נקרא רגל בפני עצמו בשביל שאינו בכל מצות הרגל שלו שהרי לולב נוהג בראשון ואינו בשאר הימים בגבולין ורגל פסח נמי חלוק במצה ומרור כדאיתמר לעיל.

How is the absence of mitzvas sukkah proof that Shmini Atzeret is its own regel?  You don't take lulav on the second day of Sukkot onward (mi'doraysa), but it is still the same holiday of Sukkos.  What's the difference between the two?

Rav Zolti in his Mishnas Yaavetz (here) has a defense of Rashi, but I would like to propose answering at least this question with a yesod R' Aharon Lichtenstein has here (parenthetically, the Asif site is really one worth putting on your radar.)  R' Aharon suggests (not enough time to go through his proofs here, so ayen sham!) that there are two types of mitzvos that we do on Yom Tov: 1) there are mitzvos that have a set time in which they have to be performed, and that time just happens to be on a specific chag; 2) there are mitzvos that define the character and nature of the chag.  

For example, R' Aharon writes that blowing shofar is not something that you do on 1 Tishrei that you could just as well do on 3 Tishrei or 5 Tishrei or any other day.  Blowing shofar is what makes 1 Tishrei into "Yom Teru'a."  It defines the day.  Without shofar, you don't have a holiday that is missing a mitzvah -- take away shofar and in effect you no longer have a holiday.  

I would suggest that the same is true of sukkah.  It's not called "Chag ha'Lulav."  It's called "Chag ha'Sukkos."  The reason is not simply because the former is done only for one day and the latter for seven.  The difference is because the former mitzvah defines the nature of the day, the character of the chag.  The latter does not -- it's just a mitzvah that happens to fall out on that particular date.  

This is why according to Rashi שאין יושבין בסוכה is proof that we are dealing with a day that has a completely different character, but the fact that mi'doraysa we don't take lulav during chol ha'moed is not indicative of anything.