[emn13]

Let's be clear this is all an extreme hypothetical, right? And there are all kind of reasons not to do this; to be sure - I don't want to claim it's somehow a good idea to do this; just that in the context of discussion at this point - i.e. given that you retain the current structure of power separation, and the current voting schemes, and decide to call for fresh elections on a failed impeachment, then that there is a case to be made to reelect the entire senate as opposed to president or house. However, that precondition is an extreme hypothetical.

The reason to favor the senate over the president is clear: if they truly believe the president still should retain political support, the president isn't the you should force into a new election.

The reason to favor the senate over the house (if forced to pick just one by this hypothetical) is that the senate is the odd one out here in electoral support, and to make it harder for them to play political games with this choice; i.e. it's in their interest to pick a stance that is supported by voters, because if they don't they'll answer to them. It's also the most impactful reelection to force, because those elections are otherwise spaced the furthest apart. Essentially: by giving the house more power and the senate less, you're inching closer towards the greater stability of a parliamentary system, and additionally it has the most impact because the senate is the least frequently reelected normally. The house really should have supremacy over both senate and president; and it's OK to check that supremacy with voters to make an unleashed house suffer the consequences (because a house that really did abuse this power is still up for reelection quite quickly, and results from the senate election would be very publicly visible).

To be clear; it's just a hypothetical. It'd be better if the senate were simply dissolved, or relegated to a more advisory role as it is elsewhere, or merged into the house, and better if presidents were elected by congress, not directly - but given the hypothetical of just one releection to force, I think there is an argument to be made that a senate relection would have the most impact on the impeachment process.

To be clear: I don't think the US president being hard to unseat is a feature; that's a design flaw. Other democracies have stable foreign policies to, even when just a simple majority in the equivalent of the house is enough to immediately unseat the equivalent of the president, with no recourse to the senate at all. In fact, other democracies seem to have a more stable foreign and domestic policy, because there's less infighting and less need for hyper-speed policy making in the rare moments without gridlock; none of that "but we can't approve that deal because currently the house or the senate aren't in the hands of the same party as the president". The decision-making capability of a parliamentary government isn't perfect, but it's less likely to gridlock for long periods of time.

It also means there's no hope of any party of trying to blockade the other should they lose, so there's more incentive to seek consensus beforehand because they simply won't be able to force anything should an election ever go the other way. And there's more potential cost for reversing previous governments policies, because what's to stop a future government from playing that same game? As it is currently, a party can risk being quite confrontational, knowing that a trifecta isn't all that likely in general. There's a good chance any controversial policy made will stick around simply because there's a good chance future governments will be gridlocked for a good while; worth risking that gamble. If governments knew that a future government could reliably overturn any policy they made, they'd be more careful choosing partisan pet projects, especially those without clear popular support.

[saberdancer]

It's hard to answer everything so I'll just focus on a few key aspects.

Wouldn't forcing senate reelection in case of failed impeachment allow the party that controls the house to force that reelection?

Let's say Party A controls the house while Party B controls the Senate and the Presidency. If Party A wants, they would be able to send impeachment to trial without real reason and then by the fact that Senate didn't convict you would get a reelection of Senate.

Way I see it, this would allow manipulation of the election process. In my mind, it would be much more logical that the House has to be reelected as they pushed forward an impeachment that was not convicted upon. This would make it so that House would not bring forward frivolous impeachments as they would have their own positions on the line.

About unseating a president, I understand your point. I agree to a degree. A sitting president can be limited in power by the fact he doesn't control the legislative branch. This may seem less optimal but it has a couple of advantages. Executive branch can keep on working within the confines allowed to them, without having to juggle for political support in the parliament. This allows the President to keep working with little regard to losing his position. This is especially good for military or foreign affairs as they can work on this with little to no political support.

[emn13]

Right, but that's a plus - that's more how it works in other democracies; the equivalent of the house having the upper hand; essentially impeachment at that point turns into a milder form of a vote of no confidence. It's still milder than in other countries, because the senate gets a vote at all.

Framing this as if any of the legal constructs (house, senate, presidency) is reasonable, and thus needs to perhaps be punished for abuse of power is I believe pointless. Impeachment is not about right or wrong, it's about political support (I mean, just look at those almost completely party-line votes in house and senate recently!)

For American democracy to start working again - which has nothing to do with fairness, mind you - the power hierarchy needs to clearly weed out obstructionism; and where obstructionism is possible or even valuable as a kind of common sense sanity-check, either side involved needs to be elected or appointed on the same electoral basis (because the other option is political powerplay, rendering the whole point of things like impeachment moot, as it is today).

Pick two: A government that can govern, or elections with differing structures (essentially ensuring various branches will fight simply as a form of party politics, not based on merit), or equal branches of government, where one (plain majority) cannot remove the other.

You can't have all three.

Aside, the idea that the presidency is stable because it's decoupled from congressional support is not born out by simple comparison to other countries. Quite the opposite: by splitting powers like that, foreign actors need to consider internal US politics to a large degree. Stability would be greater if a simple majority in the house had the ability to appoint a new president at a whim - not because whims are stable, but because that shapes the back and forth of political powerplay, and thus renders the house responsible for the presidencies errors, which is how it works in other countries. Countries don't tend to reappoint prime ministers all that often, especially if a coalition is involved, because not only is that likely to be punished by voters, it's quite hard to form coalitions if you break support like that. That's not to say that a coalition of microscopic parties would be a great idea for the US by the way; just that small steps in that direction would increase stability.