To reach a satisfactory understanding of the problem at hand, I think you need to learn about multidimensional resultants (see below for where to get started).
Working over the field $\mathbb{C}$, let $F_1(x),\ldots, F_n(x)$ be $n$ homogeneous polynomials of respective degrees $d_1,\ldots,d_n$. Then there is a polynomial in the coefficients of these forms called the resultant $\mathrm{Res}(F_1,\ldots,F_n)$ which is zero iff there exists $x\in\mathbb{C}^n\backslash\{0\}$ such that $F_1(x)=0,\ldots,F_n(x)=0$.
This resultant is multihomogeneous of degree $\prod_{j\neq i}d_j$ in the coefficients of the form $F_i$.
Now for your situation, one should consider $\mathrm{Res}(p_1,\ldots,p_{n-1},u)$
where $u(x)$ is a generic linear form. If I remember correctly, the common zero set in projective space of your $n-1$ quadratics is zero dimensional iff the above polynomial in the coefficients of $u$ does not vanish identically (with the $p$'s fixed).
Finally, if this zero dimensional condition is satisfied and you look for the coordinates of the solutions of the system along a fixed axis, these indeed are the roots of a degree $2^{n-1}$ polynomial in one variable. This follows from the above polynomial being of degree $2^{n-1}$ with respect to $u$.
A great introductory reference for multidimensional resultants is the book chapter "Introduction to residues and resultants" by Cattani and Dickenstein.
