I think given time at a blackboard we could walk through Newton's cannon in the context of Poisson gravity, and for extra credit with the cannonball inducing a perturbation of the Poisson vector field. Even without the cannonball's backreaction, the Poisson picture offers a nice image of the gravitational potential energy at the top of the cannonball's inertial (ballistic) curve. We would then consider a cosmology like our own but with a recollapse: at maximum extent there is some (quasi-)Newtonian notion of gravitational potential energy for all the galaxies, since they are at the point where they begin free-falling back into a denser configuration. It's then the usual story of relating kinetic and potential energy, and recognizing that the standard cosmological frame is close to Newtonian by design. (We also have to stop this approach when the galaxies are merging enough that radiation pressure and gas ram pressure become relevant, because the errors become astronomical).
Since we don't have a blackboard in front of us to interact with, I can suggest Alan Guth's lecture notes on Newtonian cosmology. (Guth is credited with discovering cosmic inflation.) https://web.mit.edu/8.286/www/lecn18/ln03-euf18.pdf See around eqn (3.3). You could also borrow a copy of Baumann's textbook https://www.cambridge.org/highereducation/books/cosmology/53...> which studies the Poisson equation for various spacetimes, however a static spacetime gets most of the focus.
A universe which expands forever, or which expands faster in the later universe, makes a mess of this sort of approach to calculating a gravitational potential energy.
So does any apparent recession velocity that's a large fraction of c (inducing significant redshift, whatever the recession (pseudo-)"force" might be).
However, the general idea is that there is a relationship between the kinetic energy a receding galaxy (in a system of coordinates -- a "frame" -- in which these kinematics appear) and a gravitational potential energy still occurs in a non-recollapsing universe. It's just that the potential energy climbs forever, and you get an equivalent to gravitational time dilation between galaxies at different gravitational potentials (i.e., between early-universe galaxies and higher-potential modern-times galaxies).
Accelerometers in galaxies will not show a cosmic acceleration for any galaxy; they're all really close to freely-falling (local galaxy-galaxy interactions are real -- collisions and mergers and close-calls happen -- but wash out over cosmological distances; look up "peculiar velocity" for details). Therefore we can conclude that there's no real force imposing acceleration on the galaxies. However that's also true of a cannonball in a ballistic trajectory, including one on an escape trajectory or one that enters into a stable orbit. Consequently one can draw some practical comparisons between a ballistic launch from Earth into deep space and galaxies spreading out from an initially denser early part of an expanding cosmos.
> Dark energy as energy being extracted out of the universe
No, it's just a way of thinking about whatever is driving the expansion, and that doesn't dilute away with the expansion as ordinary matter and radiation does. It's not even a "real" energy in the sense that it is only an energy in the cosmological frame, and is a frame-dependent scalar quantity, whereas in the fuller theory it's just a multiplier of the metric tensor. So it's the full relativistic metric doing the work but we absorb some of that into cosmological coordinates in the cosmological frame of reference, carving up the metric tensor into a set of vectors including an expansion vector identical at every point in spacetime.
The expansion vector can also be thought of in terms of pressure: in a collapsing cosmological frame, a pressure drives galaxies together into a denser configuration. The inverse of pressure is tension, so in an expanding cosmological frame, it's a tension that pulls galaxies apart into a sparser configuration. (The reason one uses pressure or its inverse is that the matter fields are idealized as a set of perfect fluids at rest in the cosmological frame; each such fluid has an associated density and internal pressure which evolve with the expansion or contraction of the cosmos, generally becoming less positive in the time-direction of expansion (i.e., in the future direction in a universe like ours). Another way of thinking about pressure is as a measure of isotropic inflow of energy-momentum into a point; increasing pressure at a point therefore increases the curvature at that point. Tension is an isotropic outflow, and so positive tension is repulsive as opposed to the attraction from positive pressure.)
> that explanation felt unsatisfactory to me
Hopefully the above helps a bit. Unfortunately there's only so much teaching one might do in a series of HN comments, and ultimately one probably is better served in developing some grounding in the full Einstein Field Equations / Friedmann-Lemaître equations before thinking in quasi-Newtonian ways. Going the other direction tends to lead to misunderstandings and developing false intuitions when running into situations where the quasi-Newtonian picture needs post-Newtonian correction terms.
It's cool that you have all sorts of questions. You could consider signing up for part time / non-business-hours courses in relativity at a nearby community college or the equivalent, depending on where you are, or maybe just bringing a hot lunch to a lecturer there in exchange for a quick informal tutorial. Anything like that is bound to get you to better answers than raising comments on HN threads about astrophysics in the broadest sense, as answers here are often somewhere between non-standard and unreliable.