Your comments

OK - so another case where I lack understanding of how oxyplot works. I found out that there is a dependency property on the wpf Axis base class "IsAxisVisible". Each dependent axis uses this property to determine when it is rendered. Since this is a dependency property, it can be bound to and set in a style like I was attempting.

Personally, I find this confusing since the Series objects do not have a corresponding "IsSeriesVisible" property, and use the wpf base class "FrameworkElement" property "Visibility" to determine whether they are rendered.

I would have expected from a consistency standpoint that Axis would be the same - as is evidenced by my first post - especially since Axis exposes the FrameworkElement property "Visibility".

Anyway, I am glad that the source is available, otherwise, I may not have figured this out as easily as I did.
What I found out is that if you want to hide/show a series in oxyplot via the Visible property, then the following must happen and in this order:
Each bound series must be Visible to start.
The backing collection must be populated.
plotView.InvalidatePlot must be called.

Once this happens, then it is possible to show/hide any series in the plot using the Visibility property.