No expansion does not happen at every scale. Expansion is a phenomenon that applies only when you average out unbounded systems at very large scales. By unbounded I mean systems whose kinetic energy is greater in magnitude than its potential energy.
Even galaxies themselves don't expand over time.
One question would be what do you hope to gain from a potential analogy. If you want a very down to Earth, practical and somewhat physical understanding, then the simplest and best explanation is that there are systems of objects in our universe, at very very large scales, that have a group velocity that is greater than the escape velocity needed to attract it to any other system of objects, and those systems are observed to be accelerating away from each other. No analogy is needed for this, it's just a fact presented plain and simply.
There is no theory that predicts a cause for this, but the best theory of gravity, general relativity, is a very flexible and open-ended framework that allows one to plug all kinds of different and imagined scenarios into it and see the results, even if those scenarios have no actual physical interpretation. Some people did play around with imagined results like an expanding universe, a contracting universe, an infinitely large universe, a closed but unbounded universe etc etc... General relativity doesn't predict any of these universes but it does let you explore these possibilities.
When Edwin Hubble observed that galaxies are moving away from each other then this observation was made to fit into the existing theory of general relativity. In order to take the raw observation and fit it with general relativity which interprets gravity as a purely geometric phenomenon, it did so by reframing this behavior not exactly as an intrinsic motion belonging to these large scale systems but rather as if these systems are stationary but there is more and more space filling up the universe in-between these large scale objects which gives them their apparent motion.
This is a means of reconciling the geometric view of gravity, ie. space-time, with the actual observed data. The specific technical details of how this reconciliation is performed is as OP mentioned, the FLRW metric which is here but as you said is too technical for most people to appreciate:
https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr...
You might then ask, what does this solution predict happens to atoms, or our solar system? Perhaps it predicts a very small and imperceptible expansion because other forces dominate, but nevertheless it must predict something, right? This is a tempting position, but it's not quite right.
The key reason is that the FLRW metric, which explains Hubble's observation as the literal stretching of space, literally don't make sense and can't be solved for systems like planets, solar systems, or even galaxies because it can only be used if certain requirements/preconditions are fulfilled.
These requirements are present only on the absolute largest scales where the universe looks fairly even/balanced, there is no center of mass, there is no region of the universe that is more special than any other region. In our solar system the sun is a pretty special center of mass and the solar system is not evenly balanced, same thing goes for our galaxy, and hence none of the models currently studied to describe Hubble's observations work for both the extremely large scale universe as well as for other scales.
