Just found out that is you create a viz in Tableau Desktop that uses a custom WMS server and upload it to Tableau Public, it works great!
Was just posting the same answer when you got in before me, Joe.
I think you'll find that if you download the workbook and open it with Public it can't access the WMS server, though.
We chose to not enable creating WMS-based maps in the Tableau Public Desktop product. It was a decision made in the name of streamlining out some of the advanced/specialized options that would likely be more confusing to the casual or new user.
I understand and agree with your position, and I am very thankful it does work when uploaded from Desktop, but I also wanted to alert you to the discrepancy in the online help documentation I linked to.