Meritocacy: If office/role requires specific competencies, then the canditate is either required to have appropriate credentials/experience or in some cases at least has to pass a test/exam.
This is not a joke, I think there are many decisions that should be driven by experts. One obvious one would be military officials as mentioned by Wikipedia. But where are the engineers, scientists and craftsmen in politics? Hidden away as consultants mostly?
Gov officials should be tested, according to their role, before even being considered for a role. Incompetence is dangerous and neither election nor sortition can select against it.
Election: I think there is a lot of merit in election. Being able to talk well, having confidence and being popular are all positive factors that can be very important. It's just that it becomes a farce/show too quickly and the whole election process is expensive.
Maybe there is a better way, by electing a larger pool P > N where N is the office seats, one could find a sweet spot where many people need to know, like you and find you suitable.
Sortition: Now we weeded out the incompetent and the people nobody likes. At this point it would likely be more efficient and fairer to do a random selection.
It is important that this would be the last step in any number of steps for selection.
It's a problem of information access, advertising, and news media that changes an electoral process from a meritocracy selected by the electorate into a popularity contest.
Without those factors, there's no reason that an informed, rational populace would select an unsuitable candidate. They'd use their information about the candidates to select the most meritorious option.
Instead, popularity and advertising have surprisingly become more important than suitability and merit. It's not obvious that this must happen, and I think some regulation, process improvement, and innovation in this area could change the electoral process into a more reasonable one.
It's not obvious, but I think it does follow from the concept of general election.
The problem is that being informed and rational about selecting a suitable candidate is considerably harder than we think it is:
1. Society faces a large number of extremely complex issues. Candidate X says, "we should get rid of regulation Y." Are they right? Are they wrong? To figure that out, you need to know enough about the economy, about why Y exists in the first place, how many experts agree and why, how many experts disagree and why, etc. Just for this one issue, it will take hours. If you want to make an informed choice between two competent candidates, there's no way it will take you less than a week of intensive, full time study. At this point, you should damn well be getting paid for the effort.
2. How do you evaluate the character of a candidate you've never met? What if you don't know anyone who's met them? When you hire someone, it's best if you can meet them, talk to previous employers, run background checks. How can millions of people do this for presidential candidates, each to their own satisfaction? They can't.
3. How do you negotiate with fellow voters? If I want some things and you want some things, and we both need to agree on a candidate, the candidate I would pick in my corner and the candidate you would pick in your corner may very well be different from the candidate we would agree to pick if we could negotiate directly. If Clinton and Trump voters had been able to directly negotiate with each other, is there not a possibility they would have picked neither?
4. How do you get all relevant information? Depending on what is in a candidate's tax returns or on secret recordings, a voter may decide differently. Therefore, it makes sense that the voter would be able to subpoena some of that information, including classified information. After all, if I'm picking someone to run my country, I sure would like to see any classified information that betrays poor judgment on their part. In fact, I dare say that I need this information to do my job. But obviously it is still sensitive, so it cannot be provided to millions of voters.
I don't think general elections, where everyone votes, is a good system. It's not because people are dumb. It's because it scales poorly.
A simple alternative would be to pick a random subset of the electorate (a hundred or so) to act as voters, fly them to a convention center for a week, all expenses paid plus a sizeable wage, and let them interview candidates directly, subpoena relevant classified information, and negotiate with each other. I don't think anything short of that would work. Plus, it'd be much less expensive overall.
In this scenario the President would be directly accountable to the House and would be required to answer to them. The House would have the right to investigate and replace the President at any time for any reason, which should keep them honest and focused on the job.
It was also obviously intended to empower the state as a critical entity to a federation, but all of this cements the fact that there is no perfect voting system and all come with hosts of disadvantages and representation flaws.
"Perfect is the enemy of good", as the saying goes, so let me propose a thought experiment. What if the US kept its electoral college, but each state was required to allocate its electors in proportion to the relative vote share of the parties within that state (as closely as mathematically possible)? Of course this would require a constitutional amendment (or a hack like the NPVIC) but it's interesting to consider how this would change voting patterns, campaigns, results, and the other metrics by which a voting system is judged.
The big difference is probably that a proponent of elections believes that the masses have a better idea of who is suited for that job than random selection, and proponents of sortion disagree.
Choosing a policy is easier by orders of magnitude, and yet people can only be trusted to solve the hard problem, never the easy one.
One can make a similar theoretical argument that it is harder, possibly 'orders of magnitude' so for a CEO to determine the reliability of prospective hires to be accountable for decision making, and yet they invariably do despite greater ability to take executive decisions than the public
Hum... How would that work on the real world? It would carry some merit if people could have a vote of non-confidence and put politicians out of their positions, but it is very rare to see anything like this around the world.
The fact that delegating things right is way harder than doing it oneself is well accept on management. People still delegate because doing does not scale. There is no argument against direct democracy there.
Most large firms have management teams staffed by people whose job is deciding (and owning a budget, and knowing rather than doing). And that's in businesses whose scope isn't on the scale of 'basically everything'. Their existence isn't an argument against direct democracy so much as a comprehensive refutation of the argument that informing oneself about decision makers is far more difficult than making every decision
The only reasonable voting system known to date is preferential / Condorcet voting. It's successfully applied e.g. by the Debian project.
The purpose of Democratic elections is not something like "good government". We haven't the faintest idea how to do that. What elections do for us is enable bloodless transitions of power. The useless idiots currently in power can be voted out and you can replace them with the useless idiots you want to be in power instead if you can get enough votes. Will they be any better? Well of course you hope so, but the one thing I'm sure of it is that fighting a civil war over it would be worse.
So the problem to think about isn't "Which electoral system best satisfies Arrow's conflicting criteria in my opinion as an educated person who has spent a lot of time thinking about it?" but "What ensures the people who voted for Bob not Alice will accept that Alice won?".
If you're choosing a single person like the leader of your Executive or something equally consequential democratic elections are (along with sortition) one of the least terrible of the bad options you have, and complicated mechanisms like the Schulze method you're talking about are likely to mean losers are defiant so they're probably the wrong choice compared to First Past the Post or other simple methods.
If like Debian you're basically just picking the head of a club, it's scarcely matters how you do it, Schulze seems over-complicated to me but it's just not a big deal. Having democratic elections isn't important for this purpose anyway - nobody is going to kill and die for Debian leadership otherwise, but knock yourselves out.
Things rarely suck until you start using them.