How strong are the experimental constraints on the neutrino helicity really?

It is commonly accepted that all neutrinos have left-handed helicity and all anti-neutrinos are right-handed helicity.$^1$ The experimental base comes from Goldhaber's famous experiment performed in the 1950's. Its results are consistent with only left-handed neutrinos, but the uncertainties are actually rather large. I have been searching for more recent, improved measurements of the neutrino helicity, but everything I found online were only references to Goldhaber's original experiment.

It looks to me like there were no serious attempts to reproduce or improve Goldhaber's more than 60-year old result, which I find somewhat intriguing, given that the result had a major impact on the development of the Standard Model of particle physics. Yet, it is often made to appear like the question was unambiguously settled experimentally by the Goldhaber experiment.

So my questions are: have there been any experiments which were able to improve the uncertainty (say, to the 1%-level)? Or is a better measurement simply too difficult, even with modern hardware and techniques? If there really were no lab experiments for a better value of the amount of left-handed neutrinos, are there other constraints, e.g. from cosmology? If not, then I would say that in principle right-handed neutrinos are still a possibility which should be more thoroughly tested experimentally, since it would have a major impact on particle physics.

${^1}_\textrm{For the purpose of this question I'll not consider reference frames in which the observer is faster than the (massive) neutrino.}$