/u/XenophonTheAthenian has written an excellent post addressing the myths surrounding the destruction of the Library of Alexandria, and much of his observations apply here as well. Popular belief holds that the Mongol invasions in general, and the 1258 sack of Baghdad in particular, brought an end to what had previously been a flourishing intellectual culture. But even if countless manuscripts were destroyed in the course of the Mongol conquests, this did not lead to some kind of catastrophic loss of knowledge in the Islamic World or to the end of intellectual pursuits. As you mention, there existed large centers of Islamic learning outside of the regions conquered by the Mongols, and in any case the Mongols didn't destroy all, or even most of those centers that ultimately fell under their control.

On the contrary, in certain respects the Mongol conquests led to a new burst of intellectual activity. The newly established Mongol Ilkhanid state constituted a new source of patronage for scholars looking to pursue topics of interest beyond the purview of the religious sciences. The most noteworthy example of this was the observatory of Maragha, built in what's now Iranian Azerbaijan under the patronage of Hülegü, the very same figure who led the conquest of Baghdad. The Maragha observatory's construction led to a flourishing of new astronomical investigation in the eastern Islamic world, attracting a wide variety of scholars including Nasir al-Din Tusi (d. 1274), whose works became a fundamental touchstone for all future Islamic astronomers. Scholars at Maragha put together more accurate astronomical tables than had previously been used, and proposed new models to explain the motion of the heavenly bodies that challenged those of Ptolemy. Maragha's successes inspired in the fifteenth century the creation of another observatory by the Timurid prince Ulugh Beg in what is now Uzbekistan, where astronomical tables were further refined. One of the scholars who worked at Ulugh Beg's observatory, Ali Qushji, is best-known nowadays for being the first Muslim scholar that we are aware of to have put forth mathematical models proposing the possibility of a heliocentric universe. He eventually found his way to the Ottoman Empire, where some evidence suggests that his works may have influenced Copernicus in his own heliocentric thinking.

So the Mongol conquests didn't bring an end to intellectual pursuits or lead to a loss of knowledge, in fact quite the opposite occurred, at least in some fields. The notion of a decline of Islamic intellectual life in this later period stems from an older metanarrative which frames Islamic intellectual history as playing a mediating role between the ancient world and Renaissance Europe. Having completed the process of transmitting ancient works to Europe over the course of the eleventh, twelfth, and thirteenth centuries, later Islamic intellectual history could be written off as irrelevant, because they had "passed the torch," so to speak. The decline could then be attributed either to the violence of the Mongols or to some kind of revival of Islamic conservatism, with the figure most (misleadingly) associated with this being al-Ghazali (d. 1111). The narrative then gets repeated in the Ottoman period: a supposed "golden age" in the sixteenth century, followed by centuries of obscurantism and decline. We're finding now that reality was a good deal more complicated.

When it comes to the religious sciences, however, we can observe a clear geographical break during this period. The Mongol conquests decisively shifted the core regions of Sunni Islamic scholarship from Iraq and Iran to Syria and Egypt. As you suspected, the distinction between regions that fell under Mongol rule and those that didn't was significant, although not because of the destructive nature of those conquests per se. The eastern Islamic world's new non-Muslim rulers simply had no strong interest in patronizing Islamic religious scholarship on a large scale. In contrast, being seen as good Muslim rulers became essential to the legitimacy of the new Mamluk rulers of Egypt and Syria, where the intellectual climate further benefited from the flight of scholars from the rule of the Mongols. There is some debate as to the degree to which this shift was already occurring before the Mongols arrived, but if it was, they greatly accelerated it.