Do these set systems imply a partition?
They don't necessarily imply a partition, here is a counterexample.
Each set in each $S_i$ will have an even size, while $U$ will have an odd number of elements, this already guarantees that a partition is impossible.
$U$ is the nine vertices of a cycle of length $9$, whose edges are colored with $3$ colors, in an alternating way.
With the union operation each color class generates a set system, these will be the $S_i$.
In more detail, the generating edges are 01, 34, 67 for $S_1$, 12, 45, 78 for $S_2$ and 23, 56, 80 for $S_3$.
It is easy to check that your separating condition holds.