There's a fundamental difficulty with your claim that
a mathematician can't use a term before giving its accurate definition.
Mathematical definitions are always in terms of things that are already understood. One could eliminate the use of the word "set" in developing axiomatic set theory, but you would still need to define (for example) terms such as "axiom." How would you do this? You could define the word "axiom" in terms of arithmetical concepts, but then what is the definition of an integer? Or you could define it in terms of syntactic concepts such as "symbol" and "string," but then what is the definition of a "symbol" or a "string" or a "sequence"?
If you want to do anything at all, then you have to start somewhere and take something for granted, and therefore you cannot take your principle that "a mathematician can't use a term before giving its accurate definition" literally.
Developing axiomatic set theory by using set-theoretic language might, depending on the student, be a pedagogical mistake, but it is not a logical mistake. The word "set" as it is used in the development of the theory is meant to refer to a concept that you already have a clear grasp of. The "sets" that are later introduced axiomatically are distinct from that. This is the distinction between theory and meta-theory.
There is actually an advantage to developing mathematical logic in set-theoretic terms, because it then lets you see that mathematical logic, like all other branches of mathematics, can be formalized using axiomatic set theory.
However, I agree with you that this can be confusing pedagogically. It seems that a lot of people nowadays are comfortable with taking syntactic concepts such as "symbol" and "string" and "sequence" and "rule" for granted, without demanding that these concepts be defined before they are used. Therefore one could ask for a treatment that does not refer to sets at all but that refers purely to syntax. This can still get tricky because at some point you are going to need to use some nontrivial reasoning about what happens when you manipulate strings according to syntactic rules; this will require identifying strings with integers and applying basic number-theoretic results. You might then get demands to define what integers are and questions about how you know what axioms apply to integers before you have fully developed a theory of axiomatic systems. There's no canonical way to address these demands since, as I said, something has to be taken for granted, and so I don't know that anyone has written a textbook quite like what you have in mind.
Having said that, I suggest that you try taking a look at Quine's Mathematical Logic, and in particular his section on "protosyntax." This is an attempt to develop the subject syntactically, which might be what you're looking for.
Note, by the way, that if you take this route, then some of the motivation for set theory is removed, because it is no longer apparent that set theory is really a foundation for everything. Instead, syntax becomes the foundation for everything. One can then ask if our theory of syntax is sound, and the same confusion will arise all over again, but now with "syntax" being the apparently circular concept rather than "set".
Best Answer
Hormander: The Analysis of Linear Differential operators I-IV:
Reed-Simon: Methods of Mathematical Physics I-IV
Treves: Topological Vector spaces, Distributions and Kernels
Taylor: Partial Differential Equations I-III
Taylor: Pseudodifferential Operators and Nonlinear PDEs
Gelfand-Shilov: Generalized Functions I-V