Can we use Shamir secret sharing conversely? In Shamir secret sharing scheme, we have a secret and then the shares are generated based on the secret. The question is can we generate the shares before knowing the 'secret' (which is uniformly random). Specifically, suppose $n$ users choose random numbers $r_1,r_2,\dots, r_n$, can we reconstruct a secret (random number) using $t$ of them with Shamir's technique?
No, if $n$ users have independently chosen secret random numbers $r_i$, and for any given $t\in[1,n)$, we can't use Shamir secret sharing to construct a secret dependent on $t$ out of $n$ of the $r_i$, and independent of which $r_i$ are picked. The critical issue is that what Shamir secret sharing assigns to a user is dependent on what the others get.

More generally, I think there is no function taking $t\in[1,n)$ pairs $(i,r_i)$ with distinct $i$ and producing something independent of the picks but dependent on each $r_i$. I'm trying to think of a proof, but nothing has happened yet.