To have flat spacetime you have to distort things other than spacetime -- other than space and time. Which things? Mainly the speed of light, but also there's length contraction (why? because the effects of gravity in GR are anisotropic). Thinking that the speed of light might be variable is very strange, but even in flat spacetime the speed of light is always the same locally (in all directions), just not globally (nor in all directions). So now think of the path a photon in the CMB took to get to us, and now trace it backwards: as you trace it backwards through a great void it speeds up! And as you trace its exit from a great void it slows down, but it slows down extra because now the density is higher than where we are (because the universe was denser in the past, and less dense now due to its expansion). This also means that there's different amounts of red-shifting when traversing different regions of the universe at wildly different times and differing wildly in density. There is even some blue-shifting due to traversal of great voids.
We use differences in attenuation of brightness of faraway standard type 1A supernovae candles (which we assume always have the same brightness locally at the time of the event) and their red-shift to infer acceleration of the expansion of the universe. (Assuming not much gravitational lensing we can expect attenuation of brightness of standard candles to be a simple function of distance, and then we'd expect the red-shift to agree, but if it doesn't then... if the red-shift is stronger we assume that extra red-shift to be due to the acceleration of the expansion of the universe.) But the differences in red-shift might be artifacts of the great voids that the light traversed in order to get to us. That's what this timescapes hypothesis is about! The difference in z might be explainable by great voids rather than by acceleration of the expansion of the universe -- or perhaps a combination of both even.
Flat spacetime is not actually incorrect. Einstein's field equations and its solutions (e.g., Schwarzchild's and Kerr's) are mappings between flat spacetime and curved spacetime. For example, the Schwarszchild and Kerr solutions for a single massive body (non-rotating, in the Schwarszchild case, rotating in the Kerr case) are expressed as curved spacetime second derivatives of time, space, and angles where there is an `r` variable which represents the flat spacetime distance to the center of the massive body. This is so much so that one could be forgiven for wondering if curved spacetime is not just a mathematical crutch, or a projection much like -say- a Mercator projection of a 3D map onto 2D. It sure does seem like flat spacetime is more fundamental than curved spacetime given that we have a flat spacetime `r` in the solutions.