There is no requirement for disqualification to be the result of a jury conviction
First, there is no such requirement on the plain reading of Section 3 of the 14th Amendment (and the current Supreme Court quite likes a plain reading interpretation):
... shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof.
No mention of charges or a conviction.
There have been eight disqualifications: 6 from the post-Civil War era, one from 1919, and one from 2022.
Five of the nineteenth-century cases did not involve conviction, and the information about the sixth is inconclusive. Notwithstanding, these are possibly too ancient to be persuasive today - expected standards of justice have changed.
Victor Berger was a Socialist Congressman who opposed American participation in the First World War. He continued to espouse this position after America's entry publicly and was convicted under the Espionage Act. Reelected while under indictment in 1918, Congress formed a committee to determine if he should, as a convicted felon and anti-war campaigner, be permitted to take his seat. They decided no. He was then elected again in the special election called to fill his seat, and Congress again refused to allow him to take his seat. His conviction was overturned on appeal in 1921, and he ultimately served two more terms in Congress.
Couy Griffin was a New Mexico county commissioner who participated in the January 6, 2021 attack on the Capitol. During a recorded commission meeting, he also stated that he would return to DC on Inauguration Day with his weapons. When he did, he was arrested for trespass and disorderly conduct, being convicted and acquitted respectively. Following this, Citizens for Responsibility and Ethics in Washington brought a suit asking the District Court to remove him from office under the Disqualification Clause of the Fourteenth Amendment. The court did so, and his appeal to the New Mexico Supreme Court was dismissed on procedural grounds. Note, that he was never charged nor convicted of a crime directly related to "insurrection or rebellion" - lots of people trespass and that doesn't disqualify them from Federal office.
Who decides?
The historic decisions to bar individuals have originated from all three branches of the government; some have been barred due to administrative action by governors or their superiors, others have been barred by Congress itself, and others have been banned due to court cases.
As far as I know, no one who has been disqualified has ever successfully appealed. Only those disqualified by a court have sought judicial review, and all who did so were unsuccessful. Those barred through administrative action could likely seek judicial review; decisions of Congress on the matter are likely non-judicially reviewable.
At the same time, I am unaware of any challenge against a person being unsuccessful. AFAIK, these 8 are the only times it has ever come up, and they all lost.
Due process
All eight of these were afforded due process. Due process doesn't mean you get to make your case before a jury - the right to a jury only applies when you are charged with a crime. Being disqualified from holding office is not a punitive measure.
