The history of axiomatic set theory did not proceed in the way that you suggest here. Zermelo, for example, was motivated to form his axiomatic system primarily in order to give a careful proof of his well-ordering theorem, and not to avoid the set-theoretic antinomies. If trying to avoid contradiction were our primary goal, then we would almost surely scale way back and work with much weaker axiomatic systems such as those described in Simpson's book Subsystems of Second-Order Arithmetic, which suffice for a huge fraction of mathematics. Thus your "if this axiom system does not work, let us just toy with it until we get something that looks consistent" is a straw man. I don't know of anybody who has done serious work in foundations that has taken anything even remotely resembling that attitude.
That said, you can still ask for some justification for why we should expect, say, ZFC to be consistent. Michael Greinecker has given a good answer. I would like to add one key word to his account that you may find helpful if you want to do more reading on this subject: impredicativity. Intuitively, the set-theoretic paradoxes all arise because of "self-reference" in some sense. We define something by quantifying over a set that contains the thing being defined. The intuition is that if we avoid such "impredicative" definitions, by defining new sets only in terms of sets that we have already constructed, we should block the paradoxes.
I should note, however, that ZFC is generally regarded as being impredicative, so this doesn't fully answer your question in the case of ZFC. Nevertheless, the consistency of ZFC is typically justified by the describing the so-called cumulative hierarchy of sets, which is built from the ground up, and thus is based on the same intuition that if you define things in stages with each stage building on the previous stage, then self-referential loops should not have any way of arising.
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