The Y2K problem didn't happen in the end because thousands of man-years were spent solving the problem before it happened and/or minimising the disruption that errors could cause.
It probably cost hundreds of millions of dollars, but it was a global effort that ranged from people spending a few minutes debugging a minor program to huge financial software houses rewriting and checking every line of code that they had ever produced to make sure that it wasn't going to cause problems, recompiling it, fixing the new problems, repeat and mix over months or years. I know several retired programmers, about half of them were called in to help with old software that they'd worked on going back to the 1970s, in languages that time has almost forgotten. And it worked - because people took it seriously, and devoted the resources to the problem that were needed to solve it. There were problems, nevertheless, but most of them were relatively obvious and easy to fix. We don't really know that every last one was fixed successfully, though by now it's pretty likely that everything was, or at least that the effects of any remaining errors are small enough to go unnoticed.
Hopefully work on the next problem of this sort to come along - the Binary Millennium bug due to hit in 2049 - is already well in hand. But if people bury their heads in the sand and say it'll never happen, and refuse to devote resources to solving it, then they are in for a VERY nasty shock.
