Would using perfect S-boxes in F function of an unbalanced Feistel network produce a simple and secure cipher? When I read some articles about cryptography on wikipedia, I took note of the following statements (please, correct me, if any of these are wrong):


Perfect S-box (based on bent functions) provides maximum possible
security against linear (and differential?) cryptanalysis.
The problem with using perfect S-boxes is that an S-box constructed
purely from bent functions can not be invertible.
F function in Feistel network does not have to be neither balanced
nor invertible. One could even make an extremely unbalanced Feistel
network, where one side is only one bit, and have a bent function used as F.


From these statements it seems that using perfect S-boxes in F-function of an unbalanced Feistel network is an obvious choice for a simple and secure cipher. I understand, that someone would have come up with this long ago, if it was indeed so simple, but I can not find any information regarding possible downsides of such design.
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.