OK, we found it. Another case of a missing assembly stripped by the native tool chain build. These are all three that seem to be required for RadDatePicker. The first (ItemTapCommand) is what's causing the clicks not to work, the other two are for drawing borders around the form control and inside the spinner popup.

var assembliesToInclude = new List<Assembly>
{
    typeof(Telerik.XamarinForms.DataControlsRenderer.UWP.ItemTapCommand).GetTypeInfo().Assembly,
    typeof(Telerik.XamarinForms.Primitives.RadBorder).GetTypeInfo().Assembly,
    typeof(Telerik.XamarinForms.PrimitivesRenderer.UWP.BorderRenderer).GetTypeInfo().Assembly
};
 

Xamarin.Forms.Forms.Init(e, assembliesToInclude);