Mike- That's a good start in trying to think about the Big Bang, but
there are a few mistakes in how you analyze the pie picture, and the
standard physics picture is somewhat different in fundamental ways.
Let's start with your pie picture. Let's take it a step toward
being a picture of the standard BB by saying that everybody leaves the
from the same starting point with different velocities, not at
different times. Then some of your conclusions about the relative
velocity of our neighbors to us are right but others are off a little.
In this picture where everybody picks up a fixed velocity at the start
and those velocities don't change everybody's current displacement from
us is just the time since the bang times their velocity difference from
us. What we see from our point of view is just that velocity
difference. So that leads to an even simpler effect than the one you
describe, but very much along the lines of what you were after: The
current velocity relative to us is a universal constant times the
displacement from us.
That's called the Hubble law. It's approximately true, and was the first major piece of evidence for the Big Bang.
There are some obvious small corrections to take into account all
the small-scale motions in more ore less random directions. There are
more important, deep corrections to take into account that the geometry
of our universe is given by General Relativity, not Euclid. In
addition, there are a collection of fascinating corrections due to
understood and not understood physical effects.
However, the general first-glance picture of the Hubble effect is
well captured by your pie picture, so long as you analyze it carefully.
Have a look at a nice article on the Big Bang, which is still fortunately available online (at least for the moment) at this link at Scientific American
(published on 10/22/2007)