No; what you've just shown is the notion of the Universe having a finite age has been around for a long time, not that the Big Bang Cosmology's advocates have had "over a hundred years" to explain it. None of the ideas you presented, from Moses to Newton, are capable of making specific predictions of the observed characteristics of the Universe, as has been done by the BB Cosmology.
The term, however, is a little less than a hundred years old.
Once again, you are incorrect. Hoyle, father of the Steady-State Cosmology, coined the term "Big Bang" in the 1950's as a disparagement of the alternative Cosmology.
The point being that until the early 1960's relatively few Cosmologists subscribed to the Big Bang Cosmology; most were Steady Staters. It was only after Penzias and Wilson discovered the predicted CMBR that the BB Cosmology received wide-spread acceptance. Most of the scrutiny of BB Cosmology has occurred since that time, roughly 40 years, far less time than the advocates of Relativity and Quantum Mechanics have had to defend their theories.