Have you looked at Shamir secret sharing?
For your case, it seems like all $K$ players are required to reconstruct $y$.
I think this is true because if a single player $j$ decides not to share their value $a_jx_j$, then the players would add up their values and get:
$$ \sum_{i\neq j,i=1}^K a_ix_i = y - a_jx_j$$
Since $a_jx_j$ is (hopefully) uniformly random, this gives them no information about $y$.
It looks like you've included player $i$, who knows the value $y$ directly, in the set of players. From the above, this means all players need to cooperate, including player $i$, to recover $y$. But if all players decide to cooperate, they don't need any secret shares, since player $i$ has the secret value. Instead of using a secret sharing scheme, player $i$ can send nothing at first, and then when they all agree to recover the secret value $y$, then player $i$ can just send everyone the value $y$.
Shamir secret sharing can give you a $t$-out-of-$K$ scheme, so that player $i$ can compute values $x_i$ to give to every player, such that if at least $t$ players cooperate, those players can compute values for $a_i$ so that the sum of $a_ix_i$ for all cooperating players will equal $y$.
Shamir secret sharing with $t=K$ looks very similar to what you've described, the only difference being that there is no $a_i$ and the $x_i$ are allowed to be $0$. For this scheme, you would choose uniformly random $x_i$ to for all $i$ except $i=K$. Then set
$$ x_K = y - \sum_{i=1}^{K-1}x_i$$
Then any set of $K-1$ secret values are uniformly random and independent of $y$, which basically the best security guarantee you can hope for.
From these values of $x_i$, if you want the scheme to resemble your original proposal, you could pick a random non-zero $x_i'$, and set $a_i = x_i'^{-1}x_i$. In fact, each player could do this themselves, so it will not change security. But I don't see what functionality it gives you.