I'd submit that the one-child policy of the past didn't drastically affect rural farmers, a demographic covering at least two thirds of China's population in past decades. Families residing in these rural surroundings were pretty much exempted from the one-child policy, if not circumvented entirely via loopholes to sustain fertility rate of bearing at least two children per household on average. So the lifting of two-child limit set five years ago wouldn't drastically challenge the traditional principles of ruralites. Despite influx of rural migrations to cities, those who've remained in rural outskirts (or have returned) constitute a bulk of the nation's population... still. Perhaps the new census could correct that statement if mistaken.
As for urban households... it is often said the prince/princess are shouldered by six parental figures as the pseudo nuclear family. The mother, father, and two sets of grandparents. That amounts to ample college savings for the additional emperors/empresses-to-be. Outside the core family unit, the relatives are close knitted, akin to the 15 Asian elephants that recently trekked >400km northbound toward Kunming. Cousins are often referred to as siblings, while extended relatives would live in relative close proximity. Often in the same residential complex or community in order to be near the tribal elders so to speak. Cohesiveness of filial piety cultures in the East vary significantly from that of the West, such that this Confucian formula lessens the reliance on support networks found in Western religious communities. (Btw, Buddhism is quite prevalent in Chinese culture, despite China often being labelled as atheists) Will additional immediate family members compromise the integrity of this one or two-child familial paradigm? Time will tell. But time is perceived as ticking faster for an aging Party that will celebrate their 100th anniversary in a month, so the three-child policy is worth a gamble.
Other policies we've been seeing on the news lately could supplementally be aimed in tandem at increasing the birth rate.
China has historically achieved high savings rate, which nearly doubles the savings rate of the US. In the near future these household savings may be reallocated for extra child rearing expenditures. Also the proposed new property tax laws may attempt to discourage property speculation. With hopes of shifting white elephants to liquid assets for childcare & education. Beijing's new policies targeting the tutoring industry may be implemented to reign in the after-school rat race. To lessen disposable income pressures for parents with multiple kids. We'll see.