Overlap integrals in Hückel theory
Hückel theory would still work if there would be overlap. 

In fact, it is possible to create a completely new basis by applying a transformation to the basis set by which a new set of orthogonal basis functions is created. The advantage of setting $\hat{S}$ (the overlap matrix) to $\hat{I}$ (the identity matrix) is that the eigenvectors of the matrix equation can be directly interpreted in terms of the $\pi$ orbitals in the system. 

There is the opposite view in Extended Huckel Methods whereby the assumption is that the overlap entries depend upon the off-diagonal elements. In ordinary Huckel theory, the diagonal elements are represented as 
\begin{align}
\alpha_{ii} &amp;= \langle \psi_i | \hat{H} | \psi_i \rangle \\
\hat{H} &amp;= -\frac{\hbar^2}{2}\nabla^2 + V
\end{align}

But in the extended thoery, the offdiagonal elements are not denoted by $\beta_{ij} = \delta_{ij}$ but more as

\begin{align}
\beta_{ij} &amp;= - \frac{C}{2}\Big \langle \psi_i \Big |-\frac{\hbar^2}{2}\nabla^2 + V \Big | \psi_j \Big \rangle\\
&amp;= - \frac{C}{2} \left(\alpha_{ii} + \alpha_{jj} \right)S_{ij}
\end{align}
Where $S_{ij}$ is the usual notation for the overlap between the corresponding atomic orbitals.

Here $C$ is a constant, usually taken equal to $7/4$ and has units of energy.