> The ones that require a Senate supermajority are the ones that "legislate" in areas outside of Congress's normal jurisdiction. e.g. the US Congress probably can't pass a law prohibiting states from using the death penalty, but with a 2/3 Senate vote it could sign a treaty banning it.
It isn't clear that is actually true. Yes, the 1920 case of Missouri v. Holland appears to say that treaties ratified by the Senate can bind the states in ways that Acts of Congress cannot, but a number of legal scholars think there is a decent chance that SCOTUS would overturn that precedent if the issue came before it – see for example https://doi.org/10.2307%2F1123464
Suppose that, somehow, Democrats manage to gain control of both the Presidency and a two-thirds majority in the Senate. They then use that majority to ratify the Second Optional Protocol to the International Covenant on Civil and Political Rights, and then argue that the ratification outlawed the death penalty nationwide. A retentionist state goes to SCOTUS to challenge the treaty. If we assume the current conservative SCOTUS, I think a majority would likely overturn Missouri v. Holland and rule that the treaty is unenforceable as beyond the federal government's power. However, I doubt they'd rule that the legislative act of the Senate giving advice and consent, or the executive act of depositing instruments of ratification, was unconstitutional, merely that the treaty was not legally enforceable against the states. It is worth noting such a decision would not invalidate the ratification of the protocol under international law, and the US would still have an international legal obligation to obey it (unless and until they denounced it), even though the federal government would be legally powerless (under US constitutional law) to fulfil that obligation. (See also Medellin v. Texas.)
> The Senate's consent is required to ratify certain treaties.
In legal systems which adopt the dualist approach to international law, the international act of submitting the instruments of ratification of a treaty, and the domestic legislative acts necessary to enforce it, are two different things. Although the second act normally precedes the first, there is no requirement for such an ordering under international law. And I think it is very likely that SCOTUS would consider the executive act of submitting the instruments of ratification for a treaty to be beyond its power to judicially review; SCOTUS will confine its role to deciding what the legal consequences of that act are under domestic law. It may in some cases rule the executive act legally ineffective in creating domestic legal obligations, but in doing so it is not passing judgement on the constitutionality of the executive act itself. Suppose some President decided to ratify a treaty first, and hope to get legislation implementing it through Congress second. A risky move, in that if the legislation cannot be passed, the US could be left with international legal obligations which are impossible under domestic law to fulfil. But I don't see any evidence such a risky act would be either unconstitutional under domestic law or invalid under international law.
> (Another advantage of going "up" a level is that repealing or withdrawing from a treaty is more difficult the higher you go, generally requiring a similar authority to withdraw as was used to ratify.)
The President has unilateral discretion to withdraw from any treaty, irrespective of whether it is a treaty to which the Senate gave advice and consent, a congressional-executive agreement, or a sole executive agreement. So which type is used makes no difference to the President's power to withdraw. That was the effective holding of SCOTUS in the 1979 case of Goldwater v. Carter.
Now, the President does not have unilateral discretion to repeal a congressional-executive agreement insofar as it forms part of domestic US law, and the same may be true of a treaty to which the Senate gives advice and consent. But the President's inability to repeal the domestic legal effects of the treaty doesn't make any difference to the international legal effects of withdrawal – once the withdrawal is completed, it is no longer binding on the US under international law, even if some of its provisions continue to be binding under domestic US law.