(Estimating reliability of the voter is addressed in separate section at the bottom. The rest of this answer assumes that there is a way to have it.)
What we really need is for the folks that have honed their review and moderation skills more votes to work with.
If you want to leverage skills of reliable close voters and can somehow estimate these consider "teaching" the Community ♦ user to cast votes to close. I mean, make it cast a binding vote on a question that has enough close votes (3 or 4) from voters who are sufficiently reliable.
Clear benefit of this approach is that it is strictly focused on what you want and doesn't involve risky changes that could lead to hard to control side effects. You experimented with 3CV at Programmers and with 60 reviews at SO and every time you were getting undesirable side effects, and you don't really know if any other indirect way to leverage voters experience turns good enough.
Directly "converting" voters experience into Community closes has minimal side effects and you can easily tune and tweak the criteria that triggers it, for example exclude duplicates, questions with many views, positive score etc. If needed you can set specific per-site criteria or even completely turn it off.
One difficulty here could be how to break ties on close reasons. For a start, Community could abstain of casting a vote when there is no clearly winning reason. Maybe if you choose to account for "reliability score" of the voters when resolving ties this will become a corner case.
Another thing to take care of is to ensure that Community votes do not count when you estimate reliability of the voter. This is to avoid an effect of self-fulfilling prophecy. "Community cast a vote because I am known as reliable voter and this additionally increased my reliability estimate" - do that 100, 1000, 10000 times and your estimate may seriously skew.
Speaking of suggestion that triggered this discussion, to scale close votes based on the amount of rep, I would want to share my understanding of what people wanted in this request (and many similar ones). It is way too often that I see this interpreted as people willing more questions to be closed (and subsequently translated into concerns of how to handle more closures).
The way I see it, it's really different. People who complain just want to hit close limit less (or get too close to it). They don't really want more closures, they hardly care about this.
This may sound like a subtle difference but I think it is important. If my understanding is correct then the way to address the issues raised in requests like this is to figure category of users experiencing this kind of issue and find out how system can (if it can) grant them extra close votes with minimal impact and minimal risk.
And when I say minimal impact, I mean just that. Don't try to leverage extra votes obtained this way (at least until you gain sufficient evidence that these go the way you want) - instead try to make sure that these make minimal impact. For example, if you expect or observe significant increase in closures, you can try hedging this somehow, say by making votes expire faster. Focus solely on making users happier without breaking the system while doing this.
First thing you need if you pick this way is to decide what kind of users experience the issue. Does one who bumps or gets too close to the limit 100 days in a year qualify? Does one who bumps once a week? once a month? This is important because the better you estimate them, the better are your chances to get rid of complaints with minimal impact of the change.
Next thing is to pick category to best cover the group of users you want to help. Are they mostly active reviewers, or tag badge holders, or high-rep users etc? Picking the right category and narrowing your target users is important, again, to minimize the impact of the change.
The last but not the least is to estimate how reliable are close voters in the category you picked. This is critical because this defines how you will handle the issue.
If they are generally reliable, you grant them extra votes. If they aren't... well, you decline the request.
Sorry, we would be happy to give you extra votes but our stats indicate that most users experiencing this problem tend to vote wrong. We have strong reasons to expect that giving extra votes will lead to increase of incorrectly closed questions.
How to estimate close voter reliability
I believe that there would better be a dedicated, separate discussion focused solely on that. There are just too many topics to be addressed to properly fit it here.
Is it safe to only use votes to close for simplicity or other related actions need to be taken into account - reopen, Leave Closed, Leave Open?
Is it safe to ignore deleted questions for simplicity? (I bet it isn't but put it here explicitly to make sure it isn't lost)
Is it safe to ignore expired / retracted close votes for simplicity?
Is it safe to ignore edited / reopened questions for simplicity? (at SO it probably is but what about smaller sites)
Is it safe to ignore close reasons for simplicity? Is a voter who cast 1000 "unclear" votes on 1000 questions that were closed as "too broad" reliable?
Is it OK to ignore question score for simplicity?
How to account for review audits? What about disputed audits - those that become ineligible due to human factor correction?
Should "reliability score" be displayed publicly, or only to the user, or kept internal? What about "concerns that it would encourage problematic behavior"? (fate of the accept rate suggests that these are better taken into account)