I think the argument isn’t that we’re in spitting distance of the end but that we can, if you will excuse the metaphor, recognise the distant peak of the mountain representing it.
Even extrapolating Moore’s Law’s exponential growth of x2 per 1.5 years would still take ~200 more years to reach the limits of computation — Improvements to particle accelerators aren’t anywhere near as rapid — so we have a long way to go before we reach the end of experiment even if we improve as fast as possible and have almost no surprises.
Not that I disagree with your general premise that some massive surprise could be waiting for us; I happen to think there is, and I get the impression that it’s totally unconventional in the physics community to say “at least one of GR and QM is wrong, but we don’t have any experiment that tells us which”.
