Can we apply and verify the results from CERN experiments beyond experimental settings? If we can't, how can we know the results from CERN experiments aren't just artifacts of the experimental apparatuses?
That's a good question. You're right that a random grad student in a lab can't replicate, e.g., the discovery of the Higgs boson!
However, at least for the Higgs (and I believe this is standard for most such large experiments), there were actually two different groups using two different apparatuses, both at CERN.
There was the CMS[0] detector and ATLAS[1] detector, used and analyzed by two different groups of physicists, resulting in two papers[2,3], published on the same day.
Can we apply and verify the results from CERN experiments beyond experimental settings? If we can't, how can we know the results from CERN experiments aren't just artifacts of the experimental apparatuses?