(Note: I'm breaking with my own recent practice here by using the Chinese names of Manchu nobles rather than their Manchu transliterations; this is to ease reading and followups.)
Something particularly remarkable about the Qing Empire was the relative simplicity and smoothness of succession. And I do mean relative – out of ten Qing emperors between 1644 and 1912, there were only two major succession crises: the accession of the Yongzheng Emperor in 1722 was problematic owing to a number of unresolved rival claims; that of the Tongzhi Emperor in 1861 saw the overthrow of his original regency council by Cixi and Prince Gong. However, the accessions of children were actually very common: five of the ten came to the throne aged 8 or younger (by East Asian reckoning, that is rounding up). (These were the Shunzhi (6), Kangxi (8), Tongzhi (6), Guangxu (4) and Xuantong (3) Emperors). That successions remained broadly stable and uncontroversial, particularly with the final two emperors, is testament not only to the skill of the Dowager Empress and her inner circle in ensuring long-term continuity despite some decidedly short-term emperors, but also to the endurance of political traditions established early in Qing history.
Those traditions laid out a couple of clear guidelines of succession: firstly, that the successor would be chosen by the reigning emperor during their lifetime, making their legitimacy absolutely unambiguous and, critically, incontrovertible; and secondly, that successors should ideally be chosen from the next generation of the Aisin Gioro line to the current generation on the throne. Which generation someone in the clan belonged to was marked by a shared character in the personal name of male members across that generation, so for example the Yongzheng Emperor, whose personal name was Yinzhen, had brothers named Yinti, Yinxiang and so forth. The former was a uniquely Manchu custom, while the latter was also to some extent a Confucian one.
Only twice in the course of the Qing did an emperor come to the throne without both of those conditions satisfied. The first was the aforementioned Yongzheng Emperor in 1722, whose father, the Kangxi Emperor, failed to officially designate an heir, leaving him to take the throne as his father's de facto favourite. The second was the Guangxu Emperor in 1875, who came from the same generation as the late Tongzhi Emperor (both shared the character 載 zai in their personal names – the Guangxu Emperor was 載湉 Zaitian and the Tongzhi Emperor was 載淳 Zaichun). In the former case, the Yongzheng Emperor was indeed plagued by fears of rival claimants over the course of his 12-year reign, though none ever attempted to stake such claims. Nevertheless, from this reign onwards, it became customary to clearly declare a successor somehow or another (the Yongzheng Emperor declared his heir, the Qianlong Emperor, by means of a sealed letter opened after his death.) In the latter case, a variety of claimants were considered, including at least one viable next-generation choice, Pulun, a distant descendant of the Kangxi Emperor. However, the relatively young age of the Pu generation given the brevity of the past two (the Xianfeng Emperor (Yi) had died age 30, the Tongzhi Emperor (Zai) at 19), the continuing authority of the Dowager Empresses Cixi and Ci'an, and a bit of schmoozing of imperial clan members, allowed a smooth succession for the Guangxu Emperor. Even so, it was contrived that the Guangxu Emperor was succeeding the Xianfeng Emperor, not the Tongzhi Emperor, and he was also ritually adopted by the Dowager Empresses, in order to circumvent the generational problem.
So, all that said, why, in 1908, was Puyi chosen instead of Zaifeng? The answer is simple enough – Zaifeng was, well, of the Zai generation, and for a third emperor of that generation to take the throne, 47 years since the Xianfeng Emperor died, and indeed 33 years since the first Zai emperor died, was going to be a hard sell from the standpoint of ritual propriety. Moreover, with Cixi close to dying anyway when she officially announced her choice in November 1908, there was little reason to set up a situation in which she could claim continued direct regency. As such, to pick a Zai would be both improper from a ritual standpoint, and pointless from a pragmatic one.
But in that case, why Puyi instead of, well, literally any other Pu, of whom there would have been dozens if not hundreds? To name just three, Pulun, the potential choice in 1875, was still alive, and there was also a Puwei born in 1880 who would be a viable adult candidate. Indeed, both of these were also considered besides Puyi. Besides them, there was another nephew of the Guangxu Emperor, Pugong, born in 1904, who was seemingly first in line until Puyi showed up. So what gives? Why make a 3 year-old (2 by Western reckoning) emperor?
The answer, paradoxically, was probably Zaifeng. Zaifeng inherited his elder brother's title of Prince Chun (Zaitian's adoption as the successor to the Xianfeng Emperor meant he was no longer entitled to that title), and in the post-Boxer period his career had been on a steep upward trajectory, marking him as one of Cixi's favourites. As early as May 1905, foreign correspondents in Beijing were predicting that if Zaifeng had a son, said son would be more or less guaranteed to be selected heir. A cynical take would be that Zaifeng was already under Cixi's thumb, and that she was expecting to live longer and continue to exercise control over him, but a more generous (and dare I say realistic) take would be that Zaifeng could be expected to maintain the trajectory of progressive reform that had commenced after the Boxer fiasco, even in the absence of the guiding force provided by Cixi. Depending how old Puyi would be when the Guangxu Emperor died, Zaifeng could be expected to be emperor in all but name for a good number of years.
So, to restate everything in brief: