Compactness is one of the hardest things in the undergrad curriculum to teach, I find. It’s very hard for students to grasp that the issue is not “is there a finite collection of open sets that covers K” but rather “for every collection of open sets that covers K, some finite subcollection covers K.” This year I came up with a new metaphor. I asked the students to think about how, when a professor assigns a group project, there’s always a very small subgroup of the students who does all the work. That’s sort of how compactness works! Yes, the professor assigned infinitely many open sets to the project, but actually most of them are not really contributing. And it doesn’t matter if some small set of students could do the project; what matters is that some small group among the students assigned to the project is going to end up doing it! And this seems to happen — my students nodded in agreement — no matter how the professor forms the group.
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Quomodocumque 
This is one of those properties that are so simple on their face that it’s easy to mentally gloss over that one detail on first viewing. Your story-like presentation does a good job of getting them to focus on that detail. Next step for your students is understanding how and why this very simple property is such a strong and important one.
A possible competitor, based on complexity rather than simplicity, is local connectedness. First they have to understand what a connected space is, and then understand relative topology and what a connected subset of a space is, before they can even get to the neighborhood base of open connected subsets.
Ahlfors explained a related concept by saying that a metric space is totally bounded if it can be patrolled by a finite number of arbitrarily small policemen.
This metaphor seems like a good concept for 3Blue1Brown or a similar creator.
My father’s take on (sequential) compactness was “if you dump an infinite number of points into your space, they have to pile up somewhere.” From there I was always able to guess whether a given space was or wasn’t compact, i.e., the concept was not hard to grasp.
Of course compact and sequentially compact are not exactly the same, but I feel like this is one of those places where you learn the idea first, and then you learn the way that people have found most convenient to encode it, i.e. so that proofs come out shortest. Like, is epsilon-delta the way to introduce continuity? Obviously not, but it turns out that epsilon-delta proofs work out shortly and surely, so we use that as the definition.
Not sure I like this analogy, because it gives the impression that there is a priori some fixed finite subset that will always cover, in the same way that there is a priori some fixed finite subset of “hardworking” students among the entire student body. It’s also not how groupwork is supposed to work.
My experience has been that every quantifier added to a definition beyond the first one loses 40% of the remaining students, and cute stories like this just help confused students realize that they are confused. They are still unable to resolve their confusion because that takes more registers of working memory than they have.