I think the most straightforward way of showing the value of "softer" fields is to point out that the softer fields are what mathematics is based on. I always tell my students, "step 1 of math is philosophy." If you have not philosophically determined that the entities in question obey the relevant formulas, then the mathematics is useless. Thus, the ability to use mathematics requires the softer fields, such as philosophy, to even get started.
But it even goes deeper than that. The verification of mathematics requires philosophy. How do we know that the law of non-contradiction is true? This is a philosophical question. Godel incompleteness shows that there are an infinite number of proofs which require philosophical intuition - i.e., deductive mathematics will be unable to reach them.
Philosophy also determines what sort of functions should be allowed in the descriptions of reality. Some people philosophize that reality is computable, and therefore any full description of reality should only include computable functions. Others have a more expansive view, and can include other sorts of functions in their descriptions of reality.
Even for the sciences, before you can assign numbers to anything, you have to determine what the quantifiable entities are! Thus, we have to philosophize about nature before we can even apply numbers to the question. Polanyi covers a lot of this in his book, "Science, Faith, and Society." Another book in this vein is Clouser's "Myth of Religious Neutrality," where he shows that one's starting metaphysics influences what sort of mathematics they use.
Additionally, there are questions about what classifies as a proper explanation. This is, again, a philosophical question. For information on this topic, see Keas' paper, "Systematizing the Theoretical Virtues."
Other fields do not have the certainty of mathematics, and that is often what students are searching for. However, in order to gain certainty, one must structure their thinking in such a way that makes it possible. Making sure one is doing that correctly is a philosophical notion.
Additionally, there are many things which are not quantitative, or at least not in the usual sense (usually, there are aspects of anything that can be quantified, but the core reality may be non-computational). For instance, beauty is non-quantifiable. Beauty, however, is actually more primary for living a joyful life than productivity, while quantity usually contributes more to productivity than joy.
A great quote from Niebuhr from his book, "The Irony of American History":
Yet we cannot deny the indictment that we seek a solution for practically every problem of life in quantitative terms; and are not fully aware of the limits of this approach. The constant multiplication of our high school and college enrollments has not had the effect of making us the most "intelligent" nation, whether we measure intelligence in terms of social wisdom, aesthetic discrimination, spritual serenity or any other basic human achievement. It may have mad us technically the most proficient nation, thereby proving that technical efficiency is more easily achieved in purely quantitative terms than any other value of culture....No national culture has been as assiduous as our own in trying to press the wisdom of the social and political sciences, indeed of all the humanities, into the limits of the natural sciences...the result is frequently a preoccupation with the minutiae which obscures the grand and tragic outlines of contemporary history, and offers vapid solutions for profound problems.