I care that the people at CERN can make a working LHC, not that they can communicate it to me succinctly (eg, in under 2 hours).

It's fine that it takes 150-300 hours of back-and-forth communication to transfer the ideas behind how it actually works. That seems like a reasonable amount of time to communicate a complex idea to a novice.

There's also layers of understanding:

The LHC bangs two rocks together hard to see what happens when they break.

The LHC use magnets to accelerate protons fast enough they break apart because it recreates conditions similar to the early universe.

etc.

"The LHC use magnets to accelerate protons" - That statement would really confuse someone with a basic knowledge of physics because static magnetic fields can't do work and hence can't accelerate particles.

The acceleration is actually done by electric fields produced in RF cavities.

But how do you really know if the LHC works? You need people to communicate with each other.

And do you really expect politics to raise funds for such a project if nobody can explain to them what they are doing?

I don't believe you'll get a better than "smash rocks, see what happens" understanding of what the LHC does without significant time invested in understanding.

That's okay: politicians don't actually need to know how it works, they need to know why they should build it -- which is a) something they have domain expertise in and b) a much easier question.

"To see if you understand it, you must communicate it back"

That's one option. You could also apply the knowledge and see if what you think you understand has resulted in a mental model from which you can make testable predictions that correspond to the real-world behavior of the modeled thing.

That's why there's peer review.