So we need another method. There are two basic methods that we use besides elections, although there are other options including the Athenian one. The first one is testing, and the second one is ordeal. Often we combine these.
Testing is preferred when you can identify a competency associated with success in the position, can effectively test for that competency, and don't much care about the character of the person as long as they can do the job well. This was Plato's strong preference to the Athenian option, assigning roles by lottery, which he detested.
A testing system lets you skip the associated ordeal, which also allows for quick approval of qualified candidates when alacrity is needed. For example, you might let people test out of having to go to commercial drivers' school if they can demonstrate on a practical road test that they can safely handle a big truck, and on a written test that they understand the regulations of the road and how to operate it. If they've got all that, it's enough to go along with and they could learn the rest on the job.
Ordeals are preferred when the character of the person is the first consideration, and we want to make sure that we either (a) select only people whose character we have had time to be sure of, or (b) select people only after they have gone through a character-testing-and-shaping process. There are often tests worked into the ordeals, but the real issue isn't the tests or their scores, it's suffering the process.
All sorts of organizations use ordeals. Academia has the long Ph.D. process that subjects grad students to poverty, intense stress, and years of proving that they can get along with the academic structure well enough to be accepted and approved -- in addition to the various courses, tests, the dissertation, and the need to publish in approved journals. The Civil Service inducts people and then subjects them to its human resources' continuous monitoring, and its internal bureaucratic process of selection for promotion, to identify the most compliant and obedient candidates whose character guarantees that the organization's broader purpose will be pursued by its officers. Motorcycle clubs usually require prospects to spend a year or more on a probationary status, subject to service requirements and forbidden the prestige of full membership; only after a long prospecting period can they be voted on as members. Our fire department has a six-month probationary period during which you are expected to attend meetings and trainings, and to take supporting roles on calls outside the 'hot zone.' Following that, you can be voted on as a full member -- although there are still vast amounts of training courses, practical and written tests, before you will be certified as a 'Firefighter' or 'Technical Rescuer.'
Indeed, as mentioned we often combine testing and ordeal. For example, if you want to become an Army Ranger, you will first be tested for basic qualifications; then, if you pass the tests, you'll go through the Ranger Assessment and Selection Process (RASP), which is an 8-week ordeal designed to test your character. If you pass that, there are more ordeals and further tests as you progress. The SEALs famously use 'Hell Week' as part of a difficult selection process; the Special Forces have an even more onerous selection process that entails significant service before you can even begin it.
There aren't good tests for the roles we are trying to fill. The traditional ordeals, meanwhile, serve to ensure that candidates are aligned with the organization's purpose -- which is what we don't want in a reformer. That leaves us with an alternative option, like the Athenian one; or with another sort of ordeal.
In our context, WF Buckley's '100 names in the phone book' concept is close to the Athenian approach. He doubted our institutions' ability to either test or set ordeals that would produce the right kind of people. Instead, he preferred to rely upon the common sense of Americans to assign jobs of importance -- at least in theory, and as a quip. Whether he would endorse it now, when it might violate his patrician sensibilities, is not as clear; but he endorsed the principle, once upon a time.
What is really being done is the choice of an alternative ordeal. Each of these candidates has been subjected to the ordeal of torment by the state and the very thing they are being asked to reform. That guarantees someone who understands what is bad about the organization, and thus in need of reform.
What this process may not do is select for people who understand what is good about the organization that might need to be preserved -- Chesterton's Paradox of the Wall. Maybe some of these organizations don't serve a sufficient good to justify the harms they cause now. Maybe all of them don't. That is the risk, though; it is the gamble.