Leaving all of the seats in an appointed court or board vacant, or at least, without a quorum, is not equivalent to abolishing it.
For example, the U.S. Sentencing Commission has not had a quorum, and has hence been unable to conduct business requiring a vote, since 2019, three years ago. But, the U.S. Sentencing Commission still exists. Similarly, the Federal Election Commission went many months without a quorum but it still continued to exist. The federal Merit System Protection Board has likewise experienced long stretches without a quorum - it is currently going on five years.
A President's failure to appoint U.S. Supreme Court justices could cause that institution to go temporarily dormant. But, sooner or later, a subsequent President could fill the vacancies.
The lack of successful efforts to compel the President and the U.S. Senate to take action to fill vacancies on these boards and commissions (and the historic reality that many U.S. Courts have had long standing judicial vacancies) also strongly support the conclusion that the failure of the political process to fill U.S. Supreme Court vacancies would be a non-justiciable political question.
Presumably, cases in the exclusive original jurisdiction of the U.S. Supreme Court (e.g. disputes between U.S. states), and the handful of cases in which there is a direct appeal of right to the U.S. Supreme Court (mostly decisions of three judge district court panels on election law issues), would simply be dormant awaiting a quorum so that there could be ruling on the cases, until those cases become moot.
These cases make up on the order of 2% of the U.S. Supreme Court caseload (1-3 cases a year), and rulings on stays of matters pending before the U.S. Supreme Court that have not yet been ruled upon by the full court may be made by a single U.S. Supreme Court justice even in the absence of a quorum.
In the other 98% of cases that are only subject to discretionary review by certiorari or extraordinary writ by the U.S. Supreme Court, the lower court rulings of the U.S. Court of Appeals in question, the state supreme court, or the three judge district court panel for election law issues would stand (at least until the U.S. Supreme Court regained a quorum or the cases became moot).
As it is, "in the 2019-2020 term, the court issued written opinions in the fewest cases of any term in more than a century: 53, compared with 41 during the Civil War." This is so, even though the Court has nine justices and 36 law clerks, and the most polished and complete briefing of any U.S. court, to help it write these opinions. This year's similarly small docket had 33 cases awaiting a ruling on May 5, 2022 (a 70 year high), and while it is customary for the U.S. Supreme Court to clear its docket for its operating year by July 4, nothing forbids it from simply putting off ruling on all of its remaining cases for another year or two, as it has done in a handful of very well known cases such a Brown v. Board of Education.
It isn't uncommon for cases on appeal to take years to be finally ruled upon in every available forum.
For example, the two most recent executions in Arizona involved convictions that were more than three decades old.
So, the practical effect of a Supreme Court without a quorum for an extended period of time wouldn't necessarily even cause a constitutional crisis.
