You have good questions and good insight. I don't take this as an attack at all.

I think one of the trickiest things about explaining our side is that when people try to relate to us they either compare our hourly wages, or they compare their undergrad experience with our experience as students.

I understand why both of these comparisons are being made, but the issue is more subtle.

A PhD is more like an apprenticeship than an Bachelor degree. The amount of time I spend in a classroom as a student is very limited (mostly in my first year, then maybe one seminar per year). Every year after that (2-5/6) is spent focused on research. It's like a chef or a plumber; I spend a little time in-class learning, then I start doing my own work under the supervision of my adviser.

In a Bachelor's degree, the entire focus is on classroom learning. Undergrad is more closely related to high-school than it is to a PhD. Even a Master's is closer to high-school than it is to a PhD, in the sense that most of your time is in the classroom (hence why many master's are unfunded).

I agree that you shouldn't get paid to sit in a classroom learning (wouldn't that be sweet!), but most of our time is not spent in this way it's spent on Reddit.

Your math PhD is the answer to the question: "Can you solve a problem?" The culmination of my time in my PhD is my thesis. This is a test (of sorts) to see whether or not I can answer a question that has never been answered before.

A post-doc in math answers the question "Was that a fluke? Or can they do it again?". In a post-doc we are expected to produce 2-4 papers over the course of 2-3 years. This is also a kind of test.

Both of these benchmarks serve "to sort through the oversupply of people who want research positions". In both of these cases though we are producing research as our primary role and we should be thought of as researchers.

I hope that helps. Please let me know if there's something I can expand on.