and one cannot experimentally ‘prove’ the big bang
Not bloody likely. However, one can make predictions about observations and those observations (e.g., cosmic red shift) will either be consistent with the implications of the Big Bang Theory, or not. "Consistent with" is not strong positive evidence, but it allows one to choose among alternate hypotheses. Newtonian mechanics and Special Relativity do not account for the observe bending of star light by the sun during an eclipse, but General Relativity does.