The “Visibility” setting in the Display tag only affects the visibility of an object when rendered. For more info, with the Display tag selected in the “Objects” menu, right click on its icon in the Attributes section and select “Show Help.” And then in the Content browser select “Display Tag: Tag Properties.”

I think you may be confused about the use of that tag, so the Help section should clear that up. When you write, “the field is still visible,” I think that is part of your confusion, as a Display tag has nothing to do with the display of a field. Perhaps you could explain what you mean by, “animate the visibility of the field,” so that I could better understand what you’re trying to do.
Concerning your other question about polygon objects: The Help section for Basic Properties of polygon objects does list the “Enabled” setting, but it also says that it is for “generators, deformers and primitives” (as in, polygon objects are NOT included). I don’t know why it’s no longer there, if it ever was in previous versions of C4D.