Are predicates merely convenient or are they necessary in logical language? Model theory is typically based on a formal language whose production rules and syntax are designed to formalize the deductive process, and as such typically include predicate or relation symbols. Obviously, any predicate by itself constitues a (atomic) formula, but any formula, when interpreted in a model, corresponds to a predicate as well, insofar as the formula corresponds to a definable subset of the model's universe which one could then define as the interpretation of a predicate symbol.
Neither of these correspondences are bijections, but nonetheless it seems that predicates are extraneous, so the question is - are they included in the language for the same reason that the universal quantifier is included despite being able to be defined using only the existential quantifier and negation symbol, i.e. for convenience? Or is the inclusion of predicate symbols actually important, either in model theory specifically or in the philosophical motivation as a rigorous study of the deductive process?
 A: Based on your question, I'm also assuming you're trying to omit function and constant symbols.
Without predicate symbols, there are very few definable sets. If $\mathcal{M}$ is a structure in the empty language, then: 


*

*The sets definable without parameters in $\mathcal{M}$ are precisely the emptyset and the whole domain of $\mathcal{M}$.

*The sets definable with parameters in $\mathcal{M}$ are precisely the finite and cofinite subsets of the domain of $\mathcal{M}$.
Basically, "atomic predicate symbols" are needed for the process of building a family of definable sets to even get off the ground.
Of course, given a structure $\mathcal{M}$ we can always look at it as simply a family of definable sets - that is, a pair $(M; \mathbb{F})$ where $M$ is the domain of $\mathcal{M}$ and $\mathbb{F}$ is the family of all $\mathcal{M}$-definable subsets of $M$. We can even do better, and look at the pair $$Def(\mathcal{M})=(M; \{\mathbb{F}_{\overline{a}}: \overline{a}\in\mathcal{M}^{\vert\overline{a}\vert}\})$$ where $\mathbb{F}_\overline{a}$ is the family of sets definable in $\mathcal{M}$ using the tuple $\overline{a}$ as parameters.
However, the passage from $\mathcal{M}$ to $Def(\mathcal{M})$ loses lots of information. In particular, we forget how to define homomorphisms between structures, since we can't recover how a given definable set is in fact defined. 

So while it is true that in some sense the division between "atomic predicate" and "definable set" is ad hoc, it's still necessary to $(i)$ provide some way to generate definitions, and $(ii)$ be able to tell what definition(s) correspond to a given definable set. We can do this with sufficient work while avoiding the usual atomic predicate approach, but this only creates more work for us and is in the end equivalent.
