Could impeachment really bar Trump from running again?

The House of Representatives has once again voted to impeach President Donald Trump, but he is scheduled to leave office in less than a week. Current Senate Majority Leader Mitch McConnell has said that there is not time for the Senate to try President Trump before he leaves office, so the Senate trial will have to occur after Trump is no longer president.

Article I, Section 3, Clause 7 of the Constitution of the United States says: "Judgment in Cases of Impeachment shall not extend further than to removal from office, and disqualification to hold and enjoy any Office of honor, Trust, or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment, and Punishment according to Law."

So, it seems that, since Trump will be out of office by then, the Senate's judgment will be limited to disqualifying him from "hold[ing] or enjoy[ing] an Office of honor, Trust, or Profit under the United States."

But would such a judgment prevent him from running for president again?

That question devolves to whether the presidency is an "Office ... under the United States." See Seth Barrett Tillman, "An Exchange on the Disqualification Clause: Originalism & The Scope of the Constitution's Disqualification Clause," 33 Quinnipiac L. Rev. 59 (2014).

This seems to boil down to whether the Framers, by separately naming the president, vice president and "all civil Officers of the United States" in Article 2, Section 4 as those subject to impeachment, they intended to signal that the disqualification clause does not apply to the presidency or vice presidency.

The reasoning supporting that argument centers on the notion that because the president and vice president are the executive branch, and are elected by the voters, their offices are not held "under the United States." The "Offices" referred to in the disqualification clause are therefore those of unelected officials. Thus, a judgment from the Senate disqualifying Trump from holding an "Office ... under the United States" would not bar his running for president again.

Another relevant consideration centers around the Constitution's provisions giving the House the "sole Power of Impeachment" (Article I, Section 2), and the Senate "the Sole power to try all Impeachments." (Article 2, Section 3). The late President Gerald Ford, when he was House minority leader, said: "An impeachable offense is whatever a majority of the House of Representatives considers it to be at a given moment in history," and the U.S. Supreme Court has held that the Senate's power to try impeachments of federal officials is not subject to judicial review. Nixon v. United States, 506 U.S. 224 (1993).

But what about a judgment disqualifying a former president, who was a private citizen at the time of the trial, and therefore not a sitting federal official and "liable and subject to Indictment, Trial, Judgment, and Punishment according to Law?" Would the Senate's action against such a person not be subject to judicial review? 