This is different from a project like ITER, which is also hugely expensive but could be a game changer.
I've personally beta-tested their MerlinEM and it's pretty damn cool. It's very different from other technologies that already exist (e.g. Gatan detectors). You can read out images at extremely fast rates (compared to traditional electron microscopy detectors anyway). I'm guessing that this comes from the need for CERN to read-out detectors fully for every collision. Who would have known that this would have been useful for unrelated ultrafast electron scattering experiments?
Maxwell's equations were completely theoretical (read: useless) and ahead of their time - but are fundamental to the invention of radio. Foundational research creates opportunity, it cant really be targeted.
For something like particle research, the fallout (pun not intended!) will take their time to shape anything our engineering cap availity can take advantage of.
Go on and look at it. Cross off composite particles like Antihelium-4 and antihydrogen, just look for the timeline of newly verified particles by decade. Note that each was predicted far in advance of the actual verification. (As a side question, ask "What newly predicted fundamental particles are we attempting to verify the existence of here?" You will largely hear crickets.)
Now, examine each particle and ask if we use them in industry. We have had forty-five years for the tau lepton, just as an example. We're not using it. We're not going to use it.
I majored in physics. I say this not by way of lending authority to my argument, but to highlight that I "ought" to be one of the very people slavering for more high-energy physics. But I am not. I just do not think we are going to get more interesting physics out of slamming particles together at ever nearer approaches to c.
We might get some interesting engineering out of it, but we can get that by doing other things that have more application. We cannot spend a dollar twice; let's pursue multiple approaches to fusion. Or, if one is of a pure research bent, we ought to be looking for more clever ways to break our known models, rather than just scaling our current methods up into Bigger Baryon Bashers.
What are other ways to check its predictions?
Like I said, I advocate for looking for more clever ways to look for new physics, rather than bigger ways. And you must accept the concept that at some point we may have discovered all fundamental particles that can be found in the Big Baryon Bashers and that all accelerators after that are completely wasted effort, but we'd never know that, because we would say, "Well, we didn't find anything there. Gotta build a bigger one!"
However, it does seem apropos to mention that the web itself was a spin-off of work at CERN.
I was curious about things more directly related to the actual research done at CERN.