Physics, "the beginning of time" and common sense

If we accept the result of big-bang theory that time does not indefinitely extend back in the past, how can this result be smoothly integrated with the common-sense view that for every time-instant there exist others prior to it?

That is, how can we re-school our common sense so that it accepts that the idea of "beginning of time" is not an outcome deemed necessary only because of the measuring limitations of experimental physics or the structural properties of the equations used to arrive at it?

Which leads naturally to another issue: why should the experimental approach re-school common sense rather than common sense re-school us with respect to our accepted mathematical methods?

EDIT: I have posted the same question in physics stackexchange. It was rightfully closed as "opinion based", but the following highly illuminating answer was given in the comments by John Rennie:
"no physicist I know really believes time started at the big bang. We believe some theory of quantum gravity will remove the singularity. You don’t want to take the wilder claims of popular science media too seriously. There is no shortage of common sense amongst physicists".
See here.