Lack of Bracha on Sleeping in the Succah II

In my previous post (here) I mentioned the issue of why we don't make a bracha on sleeping in the succah. There are three basic approaches to this. Rabbeinu Tam says that the ikkar yeshivas succah is by eating. The Rosh answers that one doesn't make a bracha on sleeping because one may not fall asleep (see Rosh 4:3) . Finally, the Rogotchover answers that it is not a kiyum aseh when one sleeps in the succah, rather it is just an issur aseh not to sleep in the succah.
I would like to offer an answer to this question along the lines of the Rogotchover, but perhaps slightly different. We can ask a different question. We know that part of the mitzvah of yeshivas succah is "tiyul" - to simply "hang out" in the succah. Why is there no bracha on tiyul in the succah?
The most simple answer to this would seem to be that a bracha is only made on a chiyuv aseh. One is not required to spend his time in the succah. While it may be a kiyum aseh to spend time in the succah, it is not a chiyuv, and therefore there is no bracha. Why is there no bracha on a kiyum aseh? Because in a bracha one says v'tzivanu (and He commanded us) and it can only be considered a command if there is a chiyuv. In contrast, eating in the succah is a chiyuv aseh, and therefore a bracha is recited.
According to this, one may ask why not make a bracha on sleeping in the succah? Isn't it, after all, a chiyuv to sleep in the succah, similar to eating?
We have already seen that the Rogotchover says this is not the case. It is not a chiyuv aseh to sleep in the succah, rather it is an issur aseh to sleep outside of the succah. Before we try to prove the Rogotchover's yesod, let's reframe it in our terms.
As we have suggested, tiyul in general in the succah while not a chiyuv aseh, is still considered a kiyum aseh. It seems likely that sleeping in the succah should be no worse than tiyul. In fact, it would seem to be a form of tiyul. Combining this with the Rogotchover's idea, we can say that sleeping in the succah is a kiyum aseh (meaning a kiyum of tiyul) as well as an avoidance of an issur aseh(of sleeping outside the succah). However, there is not a chiyuv aseh per se to actually sleep in the succah.
Although it is hard to logically "prove" that this is true, it seems very reasonable. Let us pose the following chakirah. We know that one must sleep in the succah on succos. However, one does not have to do tiyul in the succah on succos. Why the distinction? One could suggest that sleeping in the succah is more of a "yeshivas succah" than tiyul. But is this really logical? Actually this seems counterintuitive. If you take a look at Reshimos Shiurim in Succah (on Mitzvas Yeshiva B'succah) he discusses whether sleeping is a kum v'aseh or a shev v'al taaseh (do you go to sleep or do you fall asleep?). He seems to say it's shev v'al taaseh. Essentially, sleeping could be termed as tiyul b'shev v'al taaseh. Even if you don't go this far, it would certainly seem to be a "lower" form of tiyul. So, how could sleeping be more of a yeshiva b'succah than tiyul? It seems the opposite is true!
Rather, the reason why one must sleep in the succah would seem not to be because it is more of a yeshiva b'succah than tiyul, but rather that it is a form of tiyul that is assur outside the succah. In other words, it is an issur aseh to sleep outside the succah. However, sleep cannot be "better" than tiyul that we should say that it is a chiyuv aseh to sleep in the succah, when tiyul itself is only a kiyyum aseh.
Thus, we have proven that there is not chiyuv aseh to sleep in the succah, because if there were it would be more of a yeshiva b'succah than tiyul, which is counterintuitive.