How would Private-key Cryptography be if we use quadratic polynomials as OWF's? I am aware that one-way functions (OWF's) are the core of many of the primitives in symmetric crypto, in the sense that many cryptographic primitives in the private-key setting can be constructed from these.

On the other hand, in Multivariate Public-key Cryptography the goal is to find trapdoor functions of the form $F = (f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n))$ where each $f_i$ is a quadratic multivariate polynomial over a finite field (say $\mathbb F_2$). However, the main assumption and motivation for this is that


  A uniform function of the form $F = (f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n))$ where each $f_i$ is a quadratic multivariate polynomial over a finite field is a one-way function.


Given this and the initial discussion, we would have already a symmetric world from multivariate polynomials, without additional effort. Have someone looked at these primitives? how would they be? evaluating at polynomials is extremely efficient but, would these primitives be efficient as well? If so, why aren't these already deployed?
You might look at the stream cipher QUAD invented by Berbain, Gilbert, and Patarin.

It uses the complexity of a multivariate quadratic map as the basis for its security. A drawback of this and other multivariate quadratic map crypto systems is the space needed to store a large random quadratic map.