To answer the question of the Rosh (see previous post), the Taz 432 takes what I have called before the case-by-case approach. At any moment you can bring the mitzva of yeshivas sukkah to a close by just walking out. By bedikas chametz, one must continue the mitzva until all the chametz has been removed. The Ma'adanei Yom Tov on the Rosh challenges this distinction; he argues you can end the bedika by and simply turning the task over to someone else. Perhaps we can salvage the Taz by recasting his sevara in a more "lomdish" framework. By sukkah, the bracha is on the act of yeshivas sukkah. By bedikah, the bracha is not on the act of searching but on the kiyum mitzva of having the chametz disposed of. Whether the mitava act is performed by the ba'al habayis or his shliach amounts to the same thing - until the task is complete, the kiyum has not been achieved and the bracha has not been fulfilled.
We can now understand the dispute between the Rosh and other rishonim on two levels. One can take the approach that the point of dispute is how to understand bedikas chameitz: is the act itself the mitzva, or is it just a means to the end of the kiyum mitzva of disposing of the chameitz? Alternatively, one might understand the issue at hand is one of hilchos brachos. Where an act is a means to an end, does one recite the bracha on the act, or does one recite the bracha on the kiyum mitzva? (This second debate is discussed ny R' Chaim in the stencils with respect to placing a mezuzah on someone else's doorpost for them - is the bracha recited by the one doing the mitzva act, or by the homeowner who gets the kiyum?)
I liked the suggestion made in the comments that the machlokes between the Rosh and others is whether there can be a hefsek in yeshivas sukkah at all - I think that idea is reminiscent of a chakira whether the mitzva of sukkah is just eating the k'zayis pas in the sukkah, or the entire experience of sitting in the sukkah, including everything done there, so nothing would be a hefsek.