There was another reason for quantification in the natural sciences that didn't really become compelling until the late 19th or early 20th century: the use of dimensionality to fit quantities into a large structure of related measurements. Thus, in the standard set of physical quantities, every one is either a computable function of mass, length, and time, or a pure ratio between two quantities that do have such a function. This meant that no measurable quantity was not related to all others (a particularization of the general concept that everything in the universe was related to everything else by reduction to physical objects and structures), and that all physically important concepts were in principle derivable from those objects and structures.
The 20th century saw a triumphal rise of physical reductionism in other sciences: psychology, sociology, and economics. In principle then, the measure of money could be reduced to physical first principles, and, at least theoretically, computed from them. The end of the 20th and the beginning of the 21st century saw the rise of interest in qualitative studies in the physical sciences (e.g., topology of dynamic systems and the study of self-organization), and the realization (by some, at any rate) that money isn't a measure, it's a model in the sense that harmonic motion is a model of a pendulum: accurate in representing the behavior of some measures if others are ignored, and that there is no guarantee that there is no effect of unquantifiable qualities on the system the model represents.
If we think of money as a model, one of the primary problems of the modern world becomes easily understandable. Monetary measure is often made with no regard to certain values, which are considered either irrelevant or immeasurable. Consider the computation of cost of an item; it very rarely includes the cost of retirement of the item when its use ends. The result is that the retirement cost becomes hidden; it still must be paid, but who pays it is not computable from the model. When the need to pay the cost is brought up, the reaction is often either that such a cost is outside the purview of the model in use (i.e., that it should remain hidden) or that it in fact is not quantifiable (i.e., that it's not really a cost). The cost of dumping trash into landfill is an example in which the controversy is starting to become much more important to society. Similarly, the cost of the carbon dioxide exhaust from burning hydrocarbons is just now being debated all over the world.
To answer the three questions at the end of your post in light of the above:
Can design make decisions exclusive of the quantity of money theory?Generally, no, but other factors must be taken into account in all decisions, and money must be considered vulnerable to errors of incomplete or incorrectly estimated computations. Money is always a factor to some extent; it is a useful measure of the amount of effort and resources required to complete the object being designed, and resource allocation decisions always have to be made. In some cases money need not be a part of some lower-level decisions, but when that decision percolates up to where it affects other decisions, it becomes part of the set of tradeoffs dependent on the upper-level decision. In this, design is very similar to engineering
Can an operation create things without quantifying its future in terms of margins of profit?It certainly can, but this may not result in a good outcome, when future decisions must be made in light of the resources available after the things are created. In any case, it must always be recognized that "margin of profit" is not an exact measure, but the result of applying a (possibly inaccurate or incomplete) model, and so may not be any more useful than a qualitative factor.
Can we look at a falling rock and marvel at its destiny, rather than the quantities measured in its velocity curve? Or fire, considering how it reaches for the sun?Of course, and there are many times when we should.