Because when you allow quantification over other elements of the language you're no longer in first-order logic, but rather second- or higher-order logic, where you can quantify over predicate and relational variables, predicate-of-predicate variables, etc.
Note that what you propose is not best thought of as quantifying over WFFs: it's not typical to allow the language to talk about its own WFFs and syntax in that way. In 2nd order logic, a new set of variables is introduced, $P^n_m$ (= the $m$-th variable for an $n$-ary relation), interpreted as subsets and more generally relations of individuals that the variables $v_i$ range over.
Just as you can substitute terms for free 1st order variables, in 2nd order logic you can substitute formulas for free 2nd order variables. The rules for doing so are somewhat complicated; they're spelled out in Introduction to Mathematical Logic by Alonzo Church (first published in 1944, revised in 1956).
There are very good reasons to isolate first order logic as a thing in itself, worthy of attention and study. One is that it's sufficient to formalize traditional logic, a la Aristotle, and even the logic of relations as developed in the 19th century by Schroder and others. Another reason is that first order logic has many "nice" properties (completeness, compactness) which none of the higher-order logics enjoy.
2nd and higher order logics are indeed helpful and very expressive -- they can capture distinctions in natural language which first order logic can't. But higher order logics are very different beasts: they lack the nice mathematical properties possessed by first order logic. For example, there is no complete deductive system. They're really a form of set theory — "set theory in sheep's clothing", as Quine put it. In fact, the "theory of types", Russell & Whitehead's system that provided a foundation for mathematics in a higher order logic, predates Zermelo's set theory by a decade or so. Russell maintained that this established that mathematics is logic, a philosophy of math subsequently known as logicism; but current consensus is that Russell & Whitehead accomplished a reduction of mathematics to set theory, and not to what's commonly accepted as "logic".