Chulent: VERY good question. For two reasons. (1) R. Ilson wants his Talmidim very focused on their learning; the schedule is demanding and although this is not a typical Chareidi yeshiva, it does not engage in recreational activities during the zman like many of the other post high school yeshivos do (no tiyulim, special trips, ect.). The only organized tiyulim are one during succos bein hazmanim, one during chanukah, and two shabbatonim. I imagine he fealt the location would help avoid distractions from learning and set the Yeshivah apart from others. Also, the location is very in line with the yeshiva’s message that “we each choose our own hashkafos.” An open-minded, litvish, YU-influenced can exist in the heart of Meah Shearim a block away from Neturei Karta headquarters and three blocks from R. Elyashiv, close to the Mir, ect., and have good relations with all its neighbors with whom we share a common halacha if not a common hashkafa.
(2) The second reason is that supposedly, R. Ilson’s late wife zt”l had some family connections that she used to get a desirable building in a desirable location.