Dark energy from the large structure formation? If I understand correctly, the expansion of space is like a law of motion: the space expands if there is no force to counter it. We also know that the Universe has evolved from the even 'quantum soup' into the large scale structures, and inside those large structures gravity counters space expansion. This leads to the possibility that the accelerated expansion of space is due to the large structure formation and thus difference in the expansion rate between dense and empty space regions.
What are the arguments against this explanation for the accelerated expansion of the Universe?
 A: The expansion of space is not a law of motion per se, but rather a consequence of a law (the Einstein field equations) given appropriate initial conditions. Basically those equations imply that if the universe is already expanding, it will keep expanding - though sources of gravity can slow it down, as you suggest, and if they are distributed densely enough, they can even halt and reverse the expansion.
The hypothesis of dark energy is just a situation in which a source of energy, homogeneously filling space, reinforces that dynamic in which "if it's already expanding, it keeps expanding". Quantum fields are believed to homogeneously fill space, and they seem to have a zero-point energy due to vacuum fluctuations, so if their combined zero-point energies of all the quantum fields add up to a small positive value of the right magnitude, that would give us the dark energy.
The fact that quantum fields naturally have a vacuum energy with the right qualitative properties to be dark energy, I think is already a point against other explanations. However, at present there's no meaningful independent calculation of the magnitude of the total vacuum energy, so it's totally justified to consider other hypotheses too.
One of those is the idea that we're in a giant cosmological void - i.e. matter is distributed in a very inhomogeneous way in the universe, with regions that are perhaps billions of light-years across in which galactic clusters are very scarce, compared to other regions of space. I should emphasize that in cosmology, a "void" doesn't have to be literally empty, it can just be a region of space containing much less matter than surrounding regions. So sometimes a jargon term, "underdensity", is used (the opposite being an "overdense" region).
The rate of expansion in an underdense region is faster, and fastest at the center, precisely because there's less matter to slow it down. So if you're located near the center of a big sparse void, expansion is fastest in your own neighborhood, but is slower, the further away you look. Since in cosmology, further away means further back in time, you might think the expansion of the whole universe is speeding up. But in this scenario, the rate of expansion varies with location in space, not in time.
This scenario has been a significant alternative hypothesis for many years, and it does seem to be a version of your idea. When I look to see why cosmologists reject it, what I see are papers saying that it doesn't fit the data as well as a dark energy model. In this pop science article from late 2019, someone says a void might contribute a few percent to the apparent size of the dark energy, but couldn't realistically account for the entire effect.
However, I haven't found a clear statement of exactly which cosmological observations or data sets are most at odds with the hypothesis of a void. Competing cosmological models are judged against multiple kinds of observations, and then statistics is used to estimate how well they fit the data overall. There was a comment here ten years ago saying that, "with voids, the furthest galaxies would be more affected than closer ones; with dark energy we find the opposite: closer galaxies are going faster than expected". Hopefully someone actually working in cosmology can tell us if that's the essence of it.
