Within a paradigm there will be agreement on what the important questions are. In any field of study with a shared paradigm, there will be rough agreement about what is important. The whole field may be wrong about that answer, but it is an answerable question.
How big an agreement is there?
Across math and the hard sciences, it turns out that not only does each subfield have its dominant paradigm, but people accept the paradigms in neighboring subfields, and overall paradigms across fields. Therefore when a result like the fact that neutrinos can flip types in flight was demonstrated, it was immediately recognized as important by all physicists, even ones in fields as far away as fluid mechanics.
This shows up in journals, where each little subfield has its own constellation of journals, with the best results going into a prominent journal that is shared across several subfields. And ditto at the next level, leading to a hierarchy that culminates in the most important journal of all, Nature. Which is where the most important results in all fields are announced.
This beautiful picture completely falls apart when you get out of math and the hard sciences. The problem is not that people are not doing scientific research, it is that the field hasn't agreed on a paradigm. The question of what the right approach is and what questions are important is a subject of debate, not agreement. Which immediately renders Hamming's advice questionable. Indeed this is so much true that the picture I have described is both obvious to someone who has worked in fields where there is such consensus, and sounds impossible to someone who works in fields where there isn't.
This shows up in a variety of ways. For example in journals, each school of thought has their own ranking of journals and they don't agree. Therefore no journal can ever attain the widely acknowledged prestige of top science journals. For another example, if you talk to two random psychologists about the 10 most important research results in psychology in the last 10 years, the one's list is unlikely to contain anything recognized by the other. However if you do the same with physics, the top results will have a lot of overlap. For example right now the top 10 list for both is likely to include the verification of the Higgs Boson, and the measurement of gravitational waves.
Anyways, long story short, I understand why this researcher would question our ability to know what questions are important. However for anyone in hard science, that is surprisingly achievable.