Would using perfect S-boxes in F function of an unbalanced Feistel network produce a simple and secure cipher?
Almost perfect nonlinear permutations of $\{0,1\}^n$ are the best known designs for addressing both linear and differential cryptanalysis simultaneously. However $n$ needs to be odd. Such sboxes could be used in an SPN or feistel setup, but since one usually likes not just even length but power of two sbox bitlengths, for efficiency and implementation considerations, it is not common.

For your $(n-1)$ to one bit, very unbalanced feistel idea, the two issues I see are:


Extremely slow diffusion due to the single bit.
Bent functions are not balanced.


There are balanced quadratic functions which are almost perfect one can use, however, to address 2.

This is an active area of research.

See here, for example, for more discussion of sboxes which are permutations.