Your distinction between publicly observable and empirical inquiry generally is useful and I think gets at something precise. If empirical includes first person experience, then normativity being constitutive of inquiry is itself an empirical observation. Made from inside inquiry. About what inquiry already is before it reflects on itself.
Well said. Philosophy is approximately the “self-consciousness of rationality.” Our vocabularies swell with new words that allow us to inquiry into inquiry itself.
One eccentric way to interpret Plato’s unwritten doctrine is that claim that reality is “always qualitative.” But this “quality” is always “articulated” into a world of things that “almost repeat” as they continually manifest new “aspects” of themselves. I suggest that empirical “of course” refers to experience, and experience ( I think most will agree) is first person. Is there a manifestation of world which is not “first person” ? I don’t mean to claim that rats are thrilled at a recognition of the ontological difference. I just mean that “the presence of the world” that I can sensibly talk about is some analogue of my own, which I take to be approximately our own.
To me it seems that the normativity that constitutes human inquiry is largely communal. We inherit the problems of philosophy, most intensely where we are least aware, which makes this living past leap ahead of us, constraining what we can “hear.”
Your last line touches on a fascinating theme. When phenomenology foregrounds a phenomenon, the word “foregrounding” is appropriate because the thing made “visible” is made “visible” as having been there along, and therefore in the “background,” lurking unnoticed.