TLDR: Dark energy drives the accelerated expansion of empty space between galaxy clusters. Other effects dominate the dynamics within a given cluster.
If I can get philosophical for a moment, we must remember that every equation we write down is an approximate description of nature built from simplifying assumptions and with a well defined domain of validity. So what is the domain of validity for dark energy?
cosmology as dust in the wind
When we derive the Friedmann equations from the FLRW metric, we assume that the contents of the universe have uniform density. The matter of the universe is modeled as a uniform, non-interacting dust. In this case the dust grains are galaxy clusters. By "non-interacting" we mean the galaxy clusters just sit in place unless they are carried around by the cosmological dynamics. Just dust in the wind, man.
Dark energy fits into the Einstein field equations as a cosmological constant. It has a constant energy density.
In the past the dust grains were closer together, and the universe was matter dominated. The cosmological dynamics were driven primarily by the matter in the universe. As the universe expanded, there became more empty space between the dust grains. The matter density of the universe decreased. Eventually the matter density got down to a similar scale as the dark energy density. At this point dark energy, starts to noticeably affect the cosmological dynamics. As the universe expands more, the matter density continues to decrease, but the dark energy density stays the same, leading to the dark energy dominated cosmology we see today.
The Friedmann equations describe the dynamics of galaxy clusters. That is their domain of validity.
inside a grain of dust
If we zoom in on a single grain of dust and look inside, we'll find many galaxies. The key thing to understand is that at the scale of a single galaxy cluster, the spacetime isn't dark energy dominated. The average density of matter in a cluster is way bigger than the average density of the universe. There's just way more empty space between clusters than between galaxies within a cluster.
If we apply the same cosmological assumptions at this scale the dynamics would be different than for galaxy clusters. The increased matter density means the expansion won't happen at the same rate. The rate of expansion between clusters is larger than the rate of expansion between neighbor galaxies which is larger than the rate of expansion between stars within a galaxy.
The non-interacting assumption certainly doesn't hold within a cluster. The galaxies are not just floating on the wind of cosmology, they are interacting gravitationally and affecting each other. In this case we might have to worry about solving the gravitational $N$-body problem with a non-zero cosmological constant.
The Friedmann equations which describe cosmology are not a useful approximation of the dynamics inside a galaxy. The cosmological constant (dark energy) modifies the gravitational dynamics, but it does not drive accelerated expansion in the same way it does for the empty space between galaxy clusters.