In connection with this MO problem, I wonder whether the hypergraph in question was actually vertex-transitive. And so, as a natural variation (and, perhaps, a refinement):

If the vertex set of a vertex-transitive hypergraph $H$ can be partitioned as $V_1\cup\cdots\cup V_r$, so that every edge of $H$ contains exactly one vertex from each of the partite sets $V_i$, what reasonable conditions guarantee that $H$ possesses a perfect matching?

(As an example of a reasonable condition: $H$ is non-empty. An unreasonable condition would be that $H$ is (almost) complete in the sense that it contains the edge $\{v_1,\ldots, v_r\}$ for (almost) any $v_1\in V_1,\ldots, v_r\in V_r$.)

The case $r=2$ is easy: we are then looking at vertex-transitive bipartite graphs, and every such graph has a perfect matching by Hall's marriage theorem (provided it is non-empty). Indeed, it suffices that the graph be regular. For $r=3$ vertex-transitivity is insufficient as shows, for instance, the following construction. Let $G$ be a finite abelian group of order divisible by $2$, but not by $4$. Let $V_1,V_2,V_3$ be (disjoint) copies of $G$, and consider the hypergraph $H$ on the vertex set $V_1\cup V_2\cup V_3$ whose edges are all triples $(v_1,v_2,v_3)$ with $v_1+v_2+v_3=0$. If a perfect matching in $H$ existed, then the sum of all elements of $G$, multiplied by $3$, would be equal to $0$, which is not the case.