Did Ain Jalut save Islam, or is it overrated?

The short answer

• The western campaign. At Möngke's kurultai in 1251, the Mongols planned campaigns against two powers whose claims stood alongside their own: the Song in China and the caliphate in the west. Hülegü received the western assignment, while another brother fought the Song. The caliph's spiritual authority could be accepted under the khan, but not as an equal universal claim. The assault on the Nizari fortresses and the demand for Baghdad's submission belonged to that wider design.
• Baghdad’s weakness. The caliphate's standing and its military position were very different things. Abbasid caliphs had depended on military rulers for centuries; in Baghdad, the treasury and army were at odds, and Sunni–Shia riots divided the city. Al-Musta‘sim neither submitted in time nor paid for an effective defense. One hostile description dismissed him as “a falconer.” Nasir al-Din Tusi, the Shia scholar and adviser taken from Alamut, reportedly assured Hülegü that killing the caliph would bring no cosmic punishment. The reported assurance gives a sense of the office's spiritual weight, even as its holder lacked the means to protect it.
• Egypt’s Mamluk army. Egypt's rulers had themselves been trained for war. Many Mamluks were born on the steppe, brought into military slavery and drilled as professional cavalry before the Mamluks became Egypt's rulers. Centrally barracked, supplied from Egypt's resources and operating near their base, they brought mounted skills comparable to those of their opponents. Feudal hosts and city militias had broken before Mongol armies; Kitbuqa now met a force with a different preparation and resources behind it.
• A divided theatre. The battlefield in Syria was shaped by events far beyond it. Hülegü had taken most of his army east, and summer pasture and water constrained the horses Kitbuqa retained. The Crusaders of Acre stayed neutral but let the Mamluks pass and use their markets, even though some popes had considered alliance with the Mongols. Meanwhile Berke, the Muslim ruler of the Golden Horde, was hostile to Hülegü; the caliph's death added to the conflict and to the threat at Hülegü's rear. The Mamluks' passage north and the resources available to each side depended on these political and environmental conditions.

What it changed

Cairo and the caliphate. After Baghdad's fall, the Mamluks installed a refugee Abbasid as caliph in Cairo. The office lent religious standing to their rule, and Egypt remained under Mamluk government for roughly 250 years, until the Ottoman conquest. Baghdad was still inhabited under the Ilkhanate, but it had lost its position as Abbasid capital and its regional role diminished for centuries. The caliphate now had its home in Cairo.

The Ilkhanate takes shape. Hülegü held Persia and Iraq, along with overlordship in Anatolia; these territories formed the Ilkhanate. Mamluk resistance to the southwest and Berke's hostility soon constrained it. A generation later its rulers converted to Islam. They patronized mosques and Persian painting, and supported Rashid al-Din's world chronicle. The dynasty responsible for ending the Baghdad caliphate had become a patron of Islamic institutions.

A frontier contested. The victory did not end Ilkhan campaigns in Syria. It established a frontier that both sides would continue to contest. Farther east, Japan and Vietnam would also resist Mongol invasions. Conquest had helped the empire manage internal strains; as expansion became less certain, common campaigns became a less reliable way to settle divisions among its branches.