Isn't the date of the Big Bang a bit bogus? If I understand correctly, the date of the Big Bang is an extrapolation of acceleration of the universe's expansion through time based on the erroneous assumption that the universe is approximately 13 billion years old. The age of the universe, as far as I can tell is based on the speed of light and the fact that the furthest we could possibly see into the depths of the universe is approximately 13 billion light-years. Isn't that sort of a cat chasing his tail?  It seems that there is no valid reason for believing that the universe is really of a radius of ~13 billion years. That is just as far as we can see. If this is wrong and there is a valid reason for believing that the universe is ~ 13 billion years old, then that fact makes for a very interesting possibility.  That the expansion rate of the universe is equal to the speed of light, which leads to a very interesting suggestion that I pursued for quite awhile (until it became apparent that the suggested size of the universe is not what it seems), that the expansion rate of the universe is equal to the speed of light. Or, put another way, is it possible that the speed of light is based on the expansion rate of the universe?
I'd like to correct a couple misonceptions that might be confusing you.



  If I understand correctly, the date of the Big Bang is an extrapolation of acceleration of the universe's expansion through time based on the erroneous assumption that the universe is approximately 13 billion years old.


This is a false assumption. The observable universe is much, much larger than a sphere with a radius of 13 billion light-years. The radius is actually ~46 billion light-years (see, for example, Gott et al. (2005)).

  The age of the universe, as far as I can tell is based on the speed of light and the fact that the furthest we could possibly see into the depths of the universe is approximately 13 billion light-years.


Look at pela's answer here, if you don't want to go off Stack Exchange (or see Wikipedia for another starting point). See how the age of the universe is calculated. As an approximation,$$t_\mathrm{Uni} \sim \frac{1}{H_0} = 14\,\mathrm{billion\,years}.$$This only relies upon observations of the movements of stars.

  That the expansion rate of the universe is equal to the speed of light.


The universe is not expanding at the same rate at all distances from us. Objects further from us recede at greater speeds.