I have seen the argument that Chagatai had the middle because that way he could rule over the rest of his brothers, but this seems naive and I'd like to think that a man who conquered so much of the world had a better understanding of geopolitics than that
It's important to take note of the empire as it stood in 1227, on Chinggis Khan's death, rather than as it was in the 1250s, 60s, or 70s when it incorporated most of Eurasia. In 1227 it was much smaller (smaller of course being relative, as it still controlled from Khurasan to Manchuria). The Ilkhanate did not exist yet, and the Iranian territory was still outside of the Mongol Empire. Chagatai's territory, his ulus was essentially the southwestern corner as it then existed. Chagatai's ulus, most of the former Qara-Khitai, from Almaliq to Transoxania, was, true, not as rich as whole of China. But the Mongols did not even control the whole of North China at that point. Chagatai was actually being given, essentially, the heart of the Silk Routes. Granted, these were cities that had recently suffered terribly during the invasion of Khwarezm, but included cites like Bukhara, Samarkand, and the many cities along the Amu Darya and Syr Darya, and the lush Ferghana valley.
Much of this territory was also steppe: the valuable grazing grounds needed for Mongol herds and their horses. Since the Mongol leadership never really took to living in cities, this was also their preferred habitation zones in general, so it was also providing room for Chagatai's armies.
It is often missed too, that each branch of the family would have cities and districts in regions like China allotted to them: the Khans of the Golden Horde, Chagatai Khanate, and the Ilkhanate continued to receive tribute from these Chinese regions until quite late in the empire.
The argument that 'Chagatai was given the middle to rule over his brothers,' I have never seen before, and I think is quite silly. If Chinggis had intended that, he would have made Chagatai his successor instead of Ogedai. In 1227, Chagatai's ulus was not in the middle, it was the southwestern corner of the empire as it then existed, and a prime spot for further expansion. You can blame the house of Tolui, and Mongke's younger brother Hulegu, for the Chagatayid Khanate becoming stuck in the middle and a "miserable little ulus." But for all the complaints one can make about how Chinggis handled the succession, this is not one of them.