Particle physicist here! Actually it's pretty easy to generate intense neutrino beams. You just have to accelerate protons and smash them into a lead wall. This produces all kinds of particles, including for instance muons. Since muons are charged, you can filter and focus them into a beam using magnets, and then they rapidly decay in flight, producing a beam of neutrinos going in the same direction.
As you say, they are quite difficult to detect once produced. This is exactly what the DUNE experiment will be doing. It's a large underground detector in North Dakota that will measure properties of neutrinos produced in a beam at Fermilab near Chicago.
The difficulties mentioned in the quote from the article have little to do with neutrinos. 1000 TeV is simply an insanely high energy for any particle beam.