While I agree with you on this "hack", I also believe that if we accept that geodesic incompleteness idea is right then time for all intense and purposes has a start. At least as observers within the bubble of space-time that is causally connected.
If time/space-time existed before the BigBang is probably an unanswerable question (unless we are within a 4d black hole and we can listen to waves that perturbed the matter before the formation of our universe).
> If time/space-time existed before the BigBang is probably an unanswerable question
Why? Any sign of another Big Bang going from some other point somewhere else would indicate that "our" Big Bang might not have been the first one. It is harder to prove absence though.
Because if inflation is also correct we lost every causal connection with whatever was there before, or is so diluted that we might not be able to detect.
Another possibility could be that at the time of the BigBang the energy density was so high that everything was unified, and so when forces actually separated they "tabula rasa" anything that occupied the bubble of space-time we expanded into.
However, it could also be that the fluctuations we see in the CMB are due to perturbations of what happened before. But that possible clue is better explained by comic inflation expanding quantum fluctuations at an incredible speed that disconnected them.
That's why I said it's (probably) unanswerable. I also hope I am wrong though.
If time/space-time existed before the BigBang is probably an unanswerable question (unless we are within a 4d black hole and we can listen to waves that perturbed the matter before the formation of our universe).