Sometimes that is what happens just afterward too. Witness the Rolling Stone apology.
The Rolling Stone story was about allegations that were 2 years old so the physical evidence was gone, and you’re right - even that much time was enough to make it easy to pull the wool over the public’s eyes until specific checking of minute details revealed the deceptions. There were still records and people with knowledge of events 2 years earlier so that minute checking could happen.
When the alleged event was 40 years earlier even that kind of checking can’t happen. There is NO RISK in making a false claim that is that old.
And even with an accusation that is 2 years old, if the accuser hadn’t fabricated the whole event the basic fact-checking would have checked out and there would have been little to no way to refute the allegations. If she had really been at a party that night with a guy fitting that description, nobody would be apologizing for believing her story now. In the case of Cosby, as lonq as there are women who were actually alone with him at some point, they have NO RISK of their story ever being disproven at this late date. They have nothing to lose by making the accusations.