Note that my knowledge of both proof theory and model theory is incredibly weak. I just started learning about them using Kleene's "Mathematical logic".
If I understand it correctly then one of the tasks of logic is to be explicit about our reasoning in mathematics and to understand what reasoning is valid and which is not. Model theory in Kleene's book is described as a way to think about logic where (classically) you assume that there are some objects for which we do not care about their internal structure, but the only thing that matters is that each of them is either true or false, but not both. Also, we assume that from these objects we can construct formulas, such as "A and B", "A or B" and so forth. Then we are interested in how the truth and falsity of these formulas depend on truth and falsity of our basic objects. We define this using truth tables. Then, we can always compute the truth or falsity of formulas depending on truth and falsity of our basic objects. Then we can say that logical consequence from A to B is valid if whenever A is true then B is true. Then, in principle, we could use this as a criterion whether we do mathematics right - if every logical consequence we are making is valid then it is correct and we can use it.
Now, another point of view mentioned in the book is a proof theory. There, we do not think about the truth of the objects, but about provability. By that we mean that we choose some formulas which we call axioms and we choose deduction rules which we assume can be used in our deductions, and then we say formula is provable if there is a sequence of formulas where each is either axiom or obtained from deduction rules that use formulas that appear previously in the deduction. Note that there is no truth notion here. I am concerned here because the way we choose axioms is exactly by taking into account our interpretation. For propositional calculus, for example, we can prove that formula is valid if and only if it is provable from suitably chosen axioms. So, for me, it seems that for propositional calculus proof theory is just an alternative method obtaining formulas that are valid and obtaining logical consequences that are valid. Probably, it is more convenient and less lengthy than reason about truth values, but still, we do not gain any new insight conceptually.
But my biggest concern is when we consider stronger theories, say, those for which Godel's incompleteness theorem applies. There we can show that the truth is not the same thing as provability. If I am not mistaken, you can show that things that are provable are true but not the other way around. But, we are interested in the truth in the first place. We, therefore, see that method of proving from axioms and deductive rules can give us true formulas, but we will never be able to use this method to get all of the truths. So, why do we not forget proof theory? Why don't we just argue using model-theoretic methods like truth tables or other methods that are concerned with truth? Why is it true that, for example, modern set theory is given using axioms and deductive rules, but not by saying which statements we accept as true and then we do mathematics using logical consequences which are valid. It seems to me that by restricting ourselves by axioms and deductive rules, we restrict the number of true results we can obtain.
I hope that I am not confusing you too much and that the question makes a little bit of sense. I would appreciate any comments or advice about this topic. If you want me to be more careful with what I am trying to say, please let me know and I will edit my question accordingly.