I noticed two new thoughtful answers to this question and, by coincidence, the community team discussed this very issue yesterday. I'm not writing on behalf of the team, but rather summarizing my thoughts after our discussion.
Initially the idea was that a beta site would graduate (or be folded) after 90 days or so. In that scenario, appointing moderators pro tempore was not a particularly consequential task. If a CM happened to make a bad choice, the mistake could be corrected a few months later by an election. These days, that first moderator appointment can be of great consequence as "temporary" moderators might serve for years. Appointed moderators are almost always winners in the first election after graduation because there's a powerful incumbent effect.
Meanwhile, finding qualified and willing moderators for sites that have long been in beta turns out to be an ongoing task for the community team. I think we do a pretty good job, but it's a heavy responsibility. I'd much prefer each site's community pick their leadership than me. That said, it's very uncomfortable to think about running elections on sites with few eligible voters. Even assuming we require moderators to be reelected upon graduation, a moderator might serve for years. The more nominees and voters we have, the more likely an election will result a well-qualified moderator team.
Yearly, as-needed moderator elections
One idea that some of us on the community team considered is holding regularly scheduled elections. After the first election, incumbent moderators would have the option to continue in the position without facing reelection. But if one or more moderators decide to step down, we'd run an election to replace them. Meanwhile, if the moderator team needs extra help, we could run an election to add a member to the team. (Also, both situations could be handled at the same time.) If there is neither a need for an additional nor a replacement moderator, the election wouldn't be run.
It might not be immediately clear how this will help with the "too few people having too great an influence" problem. It turns out the most common reason we replace moderators is that they stop moderating. There are all sorts of reasons people become inactive: new job, new family responsibility, sickness, lost interest in the site/topic, and just life in general. When that happens today, we need to reach out to figure out whether a new moderator is needed. If we ran yearly elections, moderators who are no longer interested or able to do the job will just not reup and thereby trigger a replacement election.
Elections are not (yet) automated
Unfortunately, running an election is somewhat time consuming. On the front end:
A CM contacts the moderator team to find out who might be stepping down and how many slots the election could be. (This would be automated by surveying the moderator team a month or so before the scheduled election.)
Grace Note prepares a meta post asking for questions to be asked of nominees a week or so before the start of the election.
Grace Note published the nominee questionnaire at the start of the election with the most upvoted questions.
When people nominate themselves, the community team watches for joke nominations and nominees who are ineligible. We also watch for heated comments on nomination statements.
When the nomination phase ends, we make sure there's at least one more nominee than available slots. (If there are too few nominees, we extend the nomination period.)
After the election ends:
I use OpaVote to determine the winner(s).
I post an announcement on meta with the results and thank any moderators who are stepping down.
I do the actual steps to install and remove moderators.
Just about all of this could be automated. But it isn't exactly the most pressing issue for me since my steps take maybe half an hour a week. I'm not sure how much longer Grace Note's steps take or if I'm missing any steps. In any case, we couldn't keep up if all 166 sites were to have elections. So I'm going to take some time today and next week to write up a specification for automating elections. Without that piece, we can't really consider running elections on beta sites.