Disjoint compact sets in a Hausdorff space can be separated

This is a very good start, but there is a slight problem with your argument: as you change $y$, your $U$ changes as well (since $U$ is constructed in terms of $y$); you should really call it $U(y)$.

Your construction gives you an open neighborhood $W(y)$ of $y$ for each $y$; $W(y)$ is disjoint from $U(y)$. But for all you know, $W(y)$ may fail to be disjoint from $U(y')$ with $y'\neq y$.

So you really still have a bit more to go before you are done.