While water is certainly a requirement for life, a considerable number of other elements and compounds are also required, not the least are silicates, also formed by reacting with oxygen. These compounds, water and silicates, are likely the two most essential for the appearance of life - water for obvious reasons, and silicates because they are the primary component of planetary formation. Silicates are likely also essential for their unique chemical features at the molecular level in an aqueous environment, features that very likely initiated abiogenesis, and its follow-on life forms.
But many other elements are also required, and most of these are also made by either fusion (carbon, etc.), or by nucleosynthesis. With many stars burning and going out in core collapse supernova in those first few hundred million years noted in the article, you should be getting a lot of everything formed in the early years following the BB.
This all relates to how quickly star systems with stable planets for evolving life could form once their essential components had been formed and dispersed. Certainly it must be longer than 200-300 million years after the BB.
After all, it likely took the proto-Earth at least a hundred million years to form from the 'solar nebula', and probably double that to complete its interaction(s) which formed the moon before stabilizing to its present mass. From these observations alone, it would seem that the first habitable planets would have required far longer than it took for all this water to appear. And this despite the higher mass densities of all these elements and compounds in that time frame.
Anyone aware of any rational studies predicting when the first habitable planets might have formed? Perhaps in the first billion years surely seems probable.