> unintuitive

I don't think it's just you. Maybe I can help; unfortunately developing intuition without going through the mathematics even at an introductory level [as in [1]] is a hard road.

The tl;dr is that the objects are certainly moving apart; the question is whether or not the distance between them is increasing, and that depends on the coordinates in which one measures distance.

In the cosmological case (and there are others in other areas of physics) the distant astrophysical objects aren't moving spatially in one particular set of coordinates, and that set is deliberately constructed to be comoving with the moving objects. Instead, the coordinates' spatial labels themselves are time-dependent, and so absorb the expansion of space. Schematically, an astrophysical object that is at (t=5,a,b,c) in the past will be at (t=4,a,b,c), (t=3,a,b,c) and so on, and in the future will be at (t=6,a,b,c), etc., where a=const,b=const,c=const. The bookkeeping of keeping a,b,c constant for all a,b,c is rolled into the metric, a factor in the Einstein Field Equations, and so one typically hears about the "metric expansion" of the universe in cosmological contexts.

(The astrophysical objects themselves are an idealization of the largest known gravitationally-bound objects: galaxy clusters. The idealization represents them as particles of an ideal dust, to underline that the metric expansion is adiabatic).

One is perfectly free to choose non-comoving coordinates; a change of coordinates does not change the physics of the expanding universe, merely how the physics are represented in calculations. For instance, one can apply spherical coordinates with the origin on you (or some idealization thereof) at all times. Jumping up and down moves everything else in the universe in that set of coordinates. The movement of the Earth, solar system, etc., all results in distant stars moving in that set of coordinates. In those coordinates, the metric expansion is a tiny movement of distant objects in those coordinates compared to you just turning 90 degrees to your left. Distant galaxy clusters can be held still in those coordinates by flying in an aeroplane.

I say an idealization of yourself, because what are you from moment to moment? Several kilograms of gas and liquid pass through you over the course of a day. Tissue turnover happens at scales of weeks to months. You weren't alive at all a couple hundred years ago, much less several billion, so where is the coordinate origin supposed to be then?

Likewise, galaxy clusters have internal motions, radiate, have gas outflows; some collide and merge. The earliest galaxies weren't around at the time the cosmic microwave background became the surface of last scattering, but idealizing on a dust (which is a type of fluid) lets one use the comoving cosmological coordinates sensibly even that long ago. And also into the far future when the universe is so large that there will be on average only one "dust" particle per Hubble volume, and galaxies may have disintegrated into radiation or collapsed into black holes or both.

If we were to switch to spherical coordinates on some well-chosen point within our own galaxy cluster, we'd note that distant galaxies' motions are overwhelmingly radially outwards. We'd still see the (least gravitationally lensed) distant galaxies in our sky holding to a pattern: angularly-smaller, redshifted quasar spectal lines, molecular gas spectral lines, and spectral lines of younger and younger-generation/lower-metallicity stars. It'd be reasonable to think about the implications for RADAR-like round trips, and the connection with RADAR-painting e.g. an aeroplane accelerating away from you during it's take-off and climb. But we also want to think about what two other galaxy clusters would see in their sky, and an expanding universe would give each of those a highly similar view. But they might not even be able to know about the spherical set of coordinates above, and certainly would not find them useful for describing their views of the cosmos.

One might pause here for a moment to consider how an Earth-centric astronomy, a Moon-centric and a Mars-centric astronomy, would each have epicycles: different ones for Mars and Earth with respect to the other planets, and very different ones for the Moon with respect to the other planets. Presumably independent civilizations throughout our solar system would eventually come up with a sun-centric astronomy.

The comoving coordinates seem very likely to be discovered by any comparable astronomers who discover and study in detail the cosmic microwave background, and eventually figure out how to remove the 10^-3 to 10^-6 local-motion multipoles from it [2]. That's at the root of the cosmological coordinates: in them the cosmic microwave background has only tiny deviations from an ideal blackbody radiator at a temperature that decreases with time according to an expanding ideal gas law (for massless gas -- a gas of light, or a photon gas if you like). And the coordinates are useful because later large scale structures (active galaxies and so on) also end up with only small deviations from an expanding ideal gas law for a massive gas, and those deviations are straightforwardly explained by the gas being self-sticky (it clumps, unlike the radiation gas).

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[1] "Many distances" http://www.astro.ucla.edu/~wright/cosmo_02.htm

[2] T.M. Davis et al., https://arxiv.org/abs/1907.12639 in section 4 has an excellent overview of approximately this process for our civilization.