Yes I know it is handwavy and misleading. But I consider it less misleading than most attempts at visualizing it.
> Ok, how does this sketch work for a low-ellipticity eccentric orbit?
At what point in the orbit does it not work as a description of what's going on locally where the orbiting body is?
> Sure, it's a set of events in a region of the whole spacetime. If we take "Big Bang" colloquially enough to include the inflationary epoch, always assuming GR is correct, then at every point in that "Big Bang" region of the whole spacetime there is a small patch -- a subregion -- of exactly flat spacetime. However, these small patches must be small because most choices of initially-close pairs of test objects can only couple to timelike curves that wildly spread in one direction (and focus in the other).
No, there is no requirement of any region of locally flat spacetime existing. It is required (outside of singularities) that, when measuring things to first order, things are flat. However in curved space-time, the curvature can be theoretically revealed in any region, no matter how small, by measurements that are sufficiently precise to show the second order deviations from flatness that we call curvature.
> I don't know how to understand your two final sentences: how do you connect the period just before the end of inflation and the expansion history during the radiation and matter epochs?
I'm just referring to the fact that the Hubble parameter is believed to have been higher in the early universe than it is today. I'm not referring to periods such as the hypothesized inflation where the behavior is not described by GR.