It is true that most books in first order logic are written using the same sort of mathematical methods as any other area of mathematics, including basic set theory. And why not? Donald Monk's older book Mathematical Logic has a good justification of this practice in the introduction.
For foundational purposes, however, the talk of sets is a sort of "false generality". Let's use ZFC as an example.
A logic text will talk of a theory as based on a language with a set of symbols. For ZFC the language has only one symbol, $\in$, and a collection of variables. So there is nothing murky about the formulas of ZFC. If you show me a sequence of symbols, I can quickly tell you whether it is a formula of ZFC.
A logic text will talk about the axioms for a theory being an arbitrary set of formulas. But for ZFC there is a concrete algorithm that enumerates all the axioms. So, rather than being a murky "set", the axioms are very explicitly stated. Now, there are an infinite number of axioms for ZFC - and it is known that no finite set of axioms will suffice. But we could use other set theories, such as NBG, which have a finite set of axioms. That makes the collection of axioms even more explicit.
Most foundational theories are like ZFC in having a completely concrete language and a completely concrete collection of axioms. So the talk of "sets" in logic texts is not needed for these special cases. This phenomenon - that theories of foundational importance tend to be very concrete - is one of the key aspects leading to what is known as finitism in mathematics.
When we turn to the model theoretic aspects of first order logic, however, talk of sets becomes more indispensable. For example, the "completeness theorem" of first order logic shows that every consistent theory in a countable language has a countable model. Thus ZFC has a countable model. There is not any computable countable model of ZFC, however (see here). So, even for very concretely axiomatized theories, there is no way to make the completeness theorem effective. In fact, far more is known about the exact ineffectiveness of the completeness theorem. But, to address that issue, we need to talk about noncomputable sets...
Rather than shoehorning themselves into a "no sets" approach, most books on logic just let the reader who is interested separate the parts of the theory that can be done without any genuine recourse to set theory from the parts that can't.