Looking at the Expanding Universe through the lens of relativity
If you work out the equations of motion for the gravitational free-fall collapse of a clouds of particles (or galaxies), you will find that, from the point of view of any of these particles, all other particles appear to be approaching. This is true in both Newtonian theory and general relativity.

The calculation is found in numerous text books and lecture notes, e.g. Susanne Höfner's, but a qualitative argument goes like this:

From your point of view, sitting on one of the particles, a particle farther from the center of the cloud does "feel" a smaller gravitational pull from the central particles. This is, I think, the reason that you think it should accelerate more slowly. However, any particle feels the gravity from all particles that are closer to the center, so the farther you are from the center, the more particles will attract you. On the other hand, the gravity from all particles that are farther away from the center cancels out, so you don't feel those.

This is, in essence, Newton's shell theorem, and it turns out to imply that all particles reach the center at the same time. The distance between each particle is getting smaller and smaller, until the cloud collapses into an point after a a time inversely proportional to the square root of the density.

This is exactly opposite to what is observed.