Saturated sets in bipartite graph
Let's fix a maximal matching $M$. Let $Z\subseteq Y$ be the set of nodes that are not matched to nodes in $X$. We can see a node $x\in X$ belongs to a saturated set if and only if there does not exist an alternating path from $x$ to a node in $Z$, i.e., a path $xy_1x_1\cdots y_kx_kz$ where $(x_i,y_i)\in M$ and $z\in Z$ (the proof is similar to the correctness proof of your algorithm). 

So you can add directions to all edges in $E$ such that edges in $M$ have the direction from $X$ to $Y$ while edges not in $M$ have the direction from $Y$ to $X$, then the nodes in $X$ that are not reachable from any node in $Z$ make up the maximal saturated set. You can run a simple BFS to see which nodes in $X$ is reachable from nodes in $Z$. The time complexity is $O\left(\sqrt{|V|}|E|\right)$.