[somenameforme]

Impeachment is not part of the normal checks and balances. It's intended as a means to be able to remove a President who starts acting against even his own party's interests, as it requires a super-majority in the Senate, which has always been understood to mean that generally a significant chunk of your own party will need to vote against you. Consequently, literally no President has ever been convicted by the Senate, though Nixon probably would have been - and that's a good example of the scenario where impeachment is intended, as his own personal actions compromised not only the integrity of his office, but also greatly negatively affected his own party. House-only impeachments are irrelevant and inconsequential gestures when there's no chance of conviction in the Senate.

For the normal balance of powers - the legislative makes laws, the executive implements them, and the judiciary ensures the latter matches the intent of the former as well as that they remain constitutional. The legislative can undermine the judiciary or the executive by passing new laws. The executive can undermine the legislative with vetos, and the judiciary by appointments. And the judiciary can undermine the legislative by deeming the laws unconstitutional, or the executive by deeming their enforcement unlawful.

No branch is particularly superior to the others. The executive has the strength of being headed by a single person, but that is tempered by it having relatively less power than the other branches.

[estearum]

No, it's really not. The idea that POTUS is allowed to retain all Constitutional powers so long as they act in favor of their party is dismissible on its face. The founders (broadly speaking) did not anticipate a two-party system and did not build in controls against this outcome.

Impeachment is intended for exactly what it says: "high crimes and misdemeanors."

Again: all of the balances you describe only work when party loyalties do not exceed loyalties to their own branch's authorities and responsibilities. In practice what we've seen (over decades) is Congress ceding power to their own party's executive, because in practice people's political fortunes are determined by "did I make the leader of my party happy" rather than "did I retain the power of Congress."

This is EXPLICITLY counter to the intended design of the Constitution. You can read the rationale for it in Federalist 51.

This is trivially provable. POTUS (of either party, but especially in the MAGA movement) can and does threaten to primary anyone in Congress who checks him, ergo you either cede power and keep your seat, or you don't and POTUS uses his extreme control over party loyalties to replace you with someone who will cede power anyway.

The two party system (natural consequence of first-past-the-post elections) is a fundamental design flaw in our Constitution which is why it doesn't exist in any government the US has helped architect since its own inception.

All of these things are related. They're an entire web of powers, as you can read about in the Federalist Papers. The founders feared factionalism and figured it would be inevitable, but did not foresee the natural equilibrium that would be found at only two parties and all the consequent pathologies we deal with today.

[somenameforme]

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.