I agree with you on just about everything you said here. If you're arguing that my description is against the intent of the Founding Fathers then I also 100% agree there. With impeachment I am speaking of the practical effect of things, not necessarily how it was intended to function. Though I can't say I recall ever reading any political philosophy around it, so to me they remain one and the same.
And I think that segues nicely into this issue as a whole. Because the Founding Fathers were extremely averse of parties and the dangers they could pose, but this is one of the few examples where they let idealism trump reality in their philosophy. They themselves almost immediately broke down into factional parties, the first being formed by Hamilton, the author of aforementioned Federalist paper, himself! And even from that early stage it became clear that parties would become the defacto norm of society.
I'd also add that there's a bit of a paradox with things like at large proportional representation. It effectively encodes parties into the system, yet remains [relatively] diverse in practice, especially without mandates on things like the minimum vote percent. While district based FPTP has no connection to parties and ostensibly maximizes competitiveness by minimizing geographic regions a candidate needs to sway. Yet of course in practice, like you said, FPTP invariably trends towards a complete bastardization of democracy with two parties at a 50/50 equilibrium.