The trigger mechanisms aren't throwing away "data which doesn't fit" - they're throwing away data which does. That's the point - the vast majority of CERN's output is going to be extremely common events we'd expect to see all the time.

No one is setting these things up to go "hmm a particle on a totally unusual trajectory - but not where the Higgs should be so junk it".

Every experiment will be based on some sort of theory: without that, you can't design the equipment, and you wouldn't be able to interpret the data that is generated.

If by "so far have shown nothing for all their effort" you mean that no new results were found, that isn't due to the current theories being bad: in fact, it is due to them being too good: they describe the results too well and thus there is not enough difference between the current theories and current results that would require a new theory.

If we threw out 99% of data based on current theories we wouldn't have figured out that there was a problem with heliocentric until the 20th century.

The counterpoint is that if we were more eager to ditch theories as soon as they fail to explain absolutely everything in the universe, we'd have tossed Newton's laws in the early 1800s because they didn't accurately predict the orbit of Uranus.

(turned out they did, but nobody at the time knew to account for perturbations caused by the as-yet-undiscovered Neptune)

And depending on how far you want to take this, the neutrino probably would've been dismissed, too, before eventually being detected.

Which is nonsensical since we are talking about collecting data, not changing theories because of the data.

"Oh look, Uranus is doing that odd things again. Doesn't fit in with Newtonian mechanics or my current pet theory so better throw it out."