It seems that no Steiner System of the form $(2, 3, 19)$ can be represented in this fashion---11084874829 such systems do exist; see here. In particular, such a system would contain $100$ blocks; but no Grassmannian of lines in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field contains $100$ points.

However, this leaves the possibility of the following (less appealing) construction. Say a scheme $S$ is $k$-rigid for a scheme $Y$ if for any $k$ points of $Y$, $S$ embeds uniquely in $Y$ (up to automorphisms of $S$) so that it passes through all $k$ points. (This seems like a natural definition; does it already have a name?) Then we may ask for varieties $S; T\subset U$ with $S$ $p$-rigid for $U$ such that any embedding of $S$ in $U$ intersects $T$ at $q$ points, where $T$ has $r$ points.

It seems that no Steiner System of the form $(2, 3, 25)$ can be represented in this fashion---many such systems do exist; see here. In particular, such a system would contain $100$ blocks; but no Grassmannian of lines in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field contains $100$ points.

However, this leaves the possibility of the following (less appealing) construction. Say a scheme $S$ is $k$-rigid for a scheme $Y$ if for any $k$ points of $Y$, $S$ embeds uniquely in $Y$ (up to automorphisms of $S$) so that it passes through all $k$ points. (This seems like a natural definition; does it already have a name?) Then we may ask for varieties $S; T\subset U$ with $S$ $p$-rigid for $U$ such that any embedding of $S$ in $U$ intersects $T$ at $q$ points, where $T$ has $r$ points.