For example: axions, an alternative DM candidate mentioned in another comment, have seen a significant growth in attention in recent years, and the usual detector technology for axion searches is currently being refined and scaled up, from benchtop-scale, dedicated experiments to lab-scale, wide searches.
At the same time, different groups which have developed past WIMP detectors are merging to collaborate on the larger, next-generation detectors. And there is R&D and prototyping happening to create detectors which, although looking for WIMPs, are sensitive in entirely different mass ranges than those of yesteryear.
The Higgs boson has numerous experiments exclude numerous mass ranges excluded before it was finally found.
> At what point do we decide that enough has been ruled out to start looking elsewhere?
"We" don't make that decision. The various institutions who pay for these things decide, one by one, that they're going to fund some thing that sounds more promising instead.
It does kinda suck that there's something there in the universe that is perniciously difficult to see--in fact, that's how it's defined--but that is so important in the way the universe works that we can't simply ignore it. But this is the universe we're given, so this is the universe we'll run experiments on.
Indeed, relevant to what an experiment like LZ might see, there really isn't much in the way of "predictions" which can be "contradicted." What we have at this point are mechanisms to calculate the interaction rate _given at least one free parameter_. If we were to detect a non-zero rate, then we would "know" the free parameter of a single-parameter theory underlying that calculation. If we were to continue to detect a non-zero rate, then we would try to do so using different materials, and look at the time dependence of the rate (or, really, the dependence of the rate on the Earth's direction of travel in our local galaxy). That would help us choose between different theories, pin down the free parameters, and confirm that what we're seeing is consistent with "heavy stuff just sitting out in the universe."
But, from a particle physics perspective, right now there are no predictions to contradict - just an opportunity to detect something.
My impression is that many physicists would disagree with this characterization entirely, and that they're eagerly working to constrain what it is or isn't. Ruling out big classes of phenomena that could be responsible for the things we observe isn't "contradicting" the predictions in the sense implied.