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 B_dR^+. Rather I should have said that we give a universal property for B_dR^+: it is the universal pro-nilpotent thickening in solid rings. *
The two reasons why I was cheating:
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 Z is already a one-parameter deformation of F_p and that the cotangent complex of k over F_p vanishes; this is the approach discussed in section six of Bhargav's notes here: 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 B_dR^+ 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, A_inf and B_dR^+ for the p-cyclotomic extension are very explicit.
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 A_inf. So it would still be morally, if not mathematically, incorrect to claim this is a new construction of B_dR^+.