NASA uses Newtonian mechanics to send spacecraft to the outer planets and beyond. Everybody knows that Newtonian mechanics stop being accurate in certain conditions, but that does not prevent it being an incredibly useful model outside those conditions.

The best experimentally-verified cosmological models we have point to the fact that the Universe was in a hot and dense state a few microseconds after a specific point of time about 13.7 billion years ago. Extrapolating those few microseconds back in time we get an unphysical result from our current models, but that simply means those models are inaccurate if applied to the beginning of time.

Contrary to the popular belief, The Big Bang model simply does not concern itself with what happened at the exact moment of Big Bang; what it does is describe the evolution of the Universe after that, and does it well.