Are you suggesting that there is no way to determine who is knowledgeable and who is not?
What is the point of universities and schools, if that is the case?
Look, I'd prefer SOME sort of aristocracy to what we have now. But I've also learned that a degree doth not a learned person make. Maybe it's because I'm a highschool dropout. But I learned how to learn before dropping out, and that matters more than being boned up on some narrow subject. How could anyone measure that?
I don't have a degree either. But it is nevertheless obvious to me that expertise can be measured to an adequate extent as to identify a minimum set of abilities to qualify a candidate for consideration when filling a job. As there is only ever one president at a time, most suitable candidates will not get the job - but that's OK, as long as whoever does get the job is not demonstrably
unsuitable.
It is annoying to me that many jobs I could do well are closed to me because I don't have a BSc; and it is also annoying that some people who do have a BSc are demonstrably incompetent at their jobs. But that doesn't imply that requiring all applicants to have a BSc as a minimum standard before any other considerations are made is a bad idea - it is a way of filtering out the no-hopers, and that's a vital step when trying to narrow down a large pool of candidates for a single position. That is catches a few people who might have done a good job is irrelevant, as long as there are more such people left as a proportion of the pool after the filter has been applied.
There are 350,000,000 Americans. Only two end up as candidates with a serious chance of becoming president. There are already rules in place to eliminate some Americans as 'unsuitable' - Under 35s, People not natural born citizens of the US, People who have not been resident for 14 years - Obviously these restrictions eliminate some people who might make good candidates for the job, but it is better to be certain to disallow any 9 year old than it is to allow for a 34 year old who is an otherwise excellent candidate. I would propose that some minimum knowledge of facts should be required for candidates. This need not include contentious issues - just basic factual information about reality.
How can you have a president who doesn't know what the constitution says? Or the names of the heads of state of the OECD nations? Why not test at least that basic knowledge, as a minimum qualification for candidacy?
As I posted elsewhere, perhaps the Presidency shouldn't be a democratically elected post at all, being as it is a single person selected from the entire eligible population. There is no real room for compromise from the voters in selecting a single position, so inevitably most voters will get a president they didn't want.
My thought (and I welcome criticism of it) is that the POTUS should be non-partisan, chosen in a manner that makes it difficult or impossible for anyone to engineer their way into the job, and a matter of civic duty, rather than a post for which only those who deeply desire it can be selected.
One way I could see it working would be like this:
The IRS compiles a list of taxpayers, ranked by the sum total of their net taxes paid in the past four years.
Starting at the top of the list, those persons who are ineligible for the presidency under the current qualifying criteria (age, citizenship, prior service as POTUS, etc.) are struck off.
Persons who have donated to a political party, PAC, Super PAC or similar are also struck off the list. Perhaps some further tests might be applied to eliminate people who are obviously unsuitable.
When the person is found who is the highest net taxpayer who is not disqualified by the above criteria, a carload of Secret Service agents is sent to collect them, and they are installed as POTUS for the next four years, after which the process is repeated.
Any person who has a serious aversion to being selected as POTUS, can simply donate $1 to the political party of their choice, prior to the end of the tax year before the 'election', thereby disqualifying themselves.
The result should be a POTUS who is fairly randomly selected, is essentially apolitical, doesn't actively hate the idea of being president, but who hasn't actively lobbied to get there either, and who has made a success of their life to the point where they pay a fair bit of tax, without stiffing the country by using large-scale tax avoidance schemes. Of course, the separation of powers will continue, allowing Congress and the courts to keep the president's power in check; Congress will have no partisan excuse to block reasonable actions by the president, and the president would not have partisan reasons to veto reasonable legislation.
It's a half-formed thought at this stage; But it seems to me to be of merit in eliminating the petty party politics that surrounds the current holder of the executive powers. Of course, it could never happen, because the people who would lose out in the implementation of such a scheme are the same people who stand to lose the most by it. But perhaps it could be implemented after your society has finished its collapse, as part of the recovery in the post-apocalyptic wasteland.