The article notes that we have enough trouble directly detecting neutrinos in the mega-eV range, while the ones we need to measure for this purpose are in the micro-eV range. Yes it would be more informative, but so hard that we assume direct measurement is practically impossible ... but the article describes an alternative method (phase-shift) of deducing temperature from other consequences of that value.
Yes I understand all that, but direct detection is definitely not impossible. The PTOLEMY experiment at Princeton is attempting to do just that, although it's very very hard. Might still be decades away.
There are people who are trying hard to find ways to see the CNB. The primary approach is to try to do something called 'coherent neutrino scattering'. If you can find the right material/metamaterial, you're off to the races.
The number of CNB neutrinos passing through your body right now is considerable.
Direct detection would be a lot more informative. It's also hard.
There are a lot of systematic effects (foregrounds, etc.) that can matter to CMB measurements.
Any indirect glimpse is still quite valuable, as deviations from the standard plan might be apparent. Any measurement of the neutrino temperature at decoupling is interesting.
(Edit: I'd also note that the last great prediction of the Big Bang is 'inflation', something about which we know next to nothing.)
OK, so indirect gets us neutrino temperature at decoupling, and direct gets us current neutrino temperature and (I guess) feeds into neutrino mass measurements and the PMNS matrix. Anything else?
If we could get a "picture" of the CNB in all directions, it could give us hints to the structure of the Universe at two seconds after the big bang, just as the CMB gives us hints to the structure of the universe at 379,000 years after the big bang.
Any new handle on the number of neutrino-like species is interesting to dark matter searches...
If you could actually image the CNB, the implications would be comparable to the CNB. Even just a measurement of the magnitude of the dipole would be compelling.
