Modifying the formal grammar of first-order logic

Question

As it is usually stated, "if $f$ is an $n$-ary function and $t_1,\ldots,t_n$ are terms, then $f(t_1,\ldots,t_n)$ is a term" and "if $R$ is an $n$-ary relation and $t_1,\ldots,t_n$ are terms, then $R(t_1,\ldots,t_n)$ is a formula."

This conception of functions and relations seems counterproductive since 1) it means that every signature needs its own formal grammar and 2) "the language of first-order logic," rather than being a language, is actually language family. To circumvent this, I thought to find an alternative grammar which allows formulas and relations to be expressed without assigning arity (thus avoiding the need to specify production rules for functions and relations individually). This is my attempt:

<formula> ::= ¬ ( <formula> )
        | <quantifier> <variable> . <formula>
        | ( <formula> <connective> <formula> )
        | <relation> ( <csl> )

<term> ::= <constant>
        | <variable>
        | <function> ( <csl> )

<csl> ::= <term> | <csl> , <csl>

<quantifier> ::= ∃ | ∀

<constant> ::= c | <constant>*

<variable> ::= v | <variable>*

<function> ::= f | <function>*

<relation> ::= R | <relation>*

Note: connectives are not listed because each functionally complete set of connectives requires a different grammar.

The basic premise is that every theory has a [countably] infinite set of function, relation, and constant symbols but that only some of these (specifically those present in the axioms) are meaningful. For example, if we have $\forall v.\exists v^*.R^*(v,v^*)$ as an axiom, we can reasonably assert that $R^*$ is a "binary" relation. While $R^*(v,v^*,v^{**})$ is still a formula, our theory says nothing about how it ought to be evaluated.

A consequence of this is that every first-order theory ends up being incomplete, since the production rules necessarily result in statements independent of the axioms (of which, the previous is an example). I don't see this as a problem, though, because 1) any sufficiently interesting theory will contain statements neither proven nor disproven by that theory and 2) the definition of "completeness" can easily be adjusted to recapture the classical notion.

Aside from this, are there any major problems with the proposed grammar for first-order logic? Have any other attempts to "unify" first-order logic been made?