$\newcommand\BdR{B_\text{dR}}\newcommand\Ainf{A_\text{inf}}$Thank you for the question! Actually, you've caught me out.  Though I didn't realize it at the time, I was indeed cheating and should have said things more carefully.  Hopefully I can partially atone here.  Below I will describe two reasons why I was cheating, one mathematical and one moral.  But for now let me just give the take-away:

**I should not have implied that we give a new construction of $\BdR$.  Rather I should have said that we give a universal property for $\BdR$: it is the universal pro-nilpotent thickening in solid rings.**

The two reasons why I was cheating:

1.  I don't think it's actually true that the contangent complex being $F[1]$ implies, for purely formal reasons, a full universal one-parameter formal deformation.  Certainly it does give the first-order deformation.  But that only uses that the $H_1$ is $F$ and the $H_0$ is $0$, not that the others vanish.  Using the vanishing of the cotangent complex in higher degrees, you do get a good deal of knowledge about higher deformations by iteratively using transitivity triangles, and it feels like it should be saying a lot, but as far as I know it's still incomplete.  To get around this problem and actually make a full one-parameter formal deformation, one option is to work relative to an already-existing one.  So in the perfect field case, you can use that $\mathbb Z$ is already a one-parameter deformation of $\mathbb F_p$ and that the cotangent complex of $k$ over $\mathbb F_p$ vanishes; this is the approach discussed in section six of [Bhargav's notes](http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf), which is where I learned it.  In the perfectoid case, you at least see in the same way that if you construct $\BdR$ by hand for a base perfectoid, then you get it for all perfectoids over that one.  Not as snazzy as what I claimed, but it's also not nothing: for example, $\Ainf$ and $\BdR$ for the $p$-cyclotomic extension are very explicit.

2. Even if it were true that you get the construction for formal reasons from the cotangent calculation, the way I know how to get the cotangent calculation uses $\Ainf$.  So it would still be morally, if not mathematically, incorrect to claim this is a new construction of $\BdR$.