Ok, it seems that if a mask modifier is used with a subdivision surface modifier, the render and the viewport subdivisions have to be the same. I think the problem it's related to the cache of the baked simulation.
When I do that, the object renders correctly.
If you add the Decimate Modifier next to appear under Skin, I see no reason "Planar" won't satisfy you even at a default Angle Limit of 5-degrees. With Wireframe enabled in (Object Properties) Viewport Display, you can see the results immediately in Object Mode and change the Angle Limit as needed.