The technology really is not up to it, though.
To give some numbers:
- The LHC has 40M "events" (bunch of collisions) a second.
- The experiments can afford to save around 2000 of them.
This is a factor of 20k between what they collide and what they can afford to analyze. There is just no conceivable way to expand the LHC computing and storage by a factor of 20k.
Valid question would be why they don't just collide fewer protons. The problem is that when you study processes on a length scale smaller than a proton, you really can't control when they happen. You just have to smash a lot and catch the interesting collisions.
So yeah, it's a lot of "throwing away data" in the smartest way possible.
-------------------
All that said, it might be a stretch to say the data is "thrown away", since that implies that it was ever acquired. The data that doesn't get saved generally doesn't make it off a memory buffer on a sensor deep within the detector. It's never piped through an actual CPU or assembled into any meaningful unit with the millions of other readouts.
If keeping the data was one more trivial step, the experiments would keep it. As it is they need to be smart about where the attention goes. And they are! The data is "thrown away" in the sense that an astronomy experiment throws away data by turning off during the day.