Edit: Although plenty of scientists do study cultural knowledge as a place to formulate potentially useful hypotheses to be tested. Ethnopharmacology is a whole field dedicated to the prospect, just for one example.
This is an excellent standard for particle physics, but can be extremely limiting in other contexts. You probably claim to "know" a great number of things, despite the fact that you have never "measured" any of these things (or could ever hope to measure them). Whether or not we can consider this knowledge is an old debate[1].
When taken to extremes, the scientific world view just devolves into "scientism", which denies the possibility that we can know _anything_ outside the strictures of the scientific method.
[1] https://plato.stanford.edu/entries/rationalism-empiricism/
Also, virtually every branch of science utilizes both quantitative and qualitative tools. I'm not sure if you meant to be dismissive of the often subtle methods for reconciling these different scientific tools, but it seemed to me that you did.
The Roman Catholic Church says that Transubstantiation is a real phenomenon. When a priest says the magic words, ordinary wine and bread are transformed into the blood and flesh of Jesus Christ. It's not a metaphor, it really happens, it's a miracle. BUT although they are flesh and blood, they continue to have all the same detectable properties they had before, there is no way to distinguish them from ordinary bread and wine.
This is a rock solid, but completely useless epistemological argument. You can do it (to similar lack of actual effect) for anything, maybe I believe that this rock is actually the entire planet Jupiter except that it has all the same properties as an ordinary rock.