Not sure if what I will now say is correct at all, but still want to say it.

Maybe in fact at least in the area of foundations the procedure-based, as you call it, approach is inevitable. Because of incompleteness already when dealing with Peano Arithmetic, and much more so in Set Theory, we are in a position of approximating our object of study (like the structure of all natural numbers with certain amount of usual operations and induction principles or, say, some cumulative hierarchy of sets) by means of gradually adding new axioms and studying their consequences, never knowing at each given step whether we have added something too strong and reached contradiction. We know that we are doomed to do it again and again, never reaching the complete description of the structure in question.

I would say this kind of approximation process is a combination of the axiomatic method with the procedures that you mention, and presence of the latter seems unavoidable.