Why was the winner of the AES competition not a Feistel cipher?
DES actually demonstrated that a Feistel structure was not a guarantee against attacks. In "academic" terms, DES is broken by both differential and linear cryptanalysis, because they require, respectively, $2^{47}$ chosen plaintexts and $2^{43}$ known plaintexts, whereas the DES key is (effectively) 56 bits. Of course, for practical attacks, we would brute force the key: computing the function $2^k$ times is vastly easier than obtaining $2^k$ known plaintext/ciphertext pairs (or, even worse, chosen plaintext/ciphertext pairs). But in the usual "academic" evaluation of security, both linear and differential cryptanalysis count as breaks.

Luby and Rackoff have demonstrated in 1988 that given "perfect" round functions, a four-round Feistel structure is secure. However, this proof has two practical issues:


It is relative to the output size of the round function, i.e. 32 bits for a 64-bit block cipher. For 128-bit security, blocks have to be 256-bit wide for the proof to actually apply; but the AES call for candidates requested 256-bit security with 128-bit blocks, not the other way round.
DES has amply demonstrated that concrete round functions cannot be assumed to be perfect.


So while the security provided by a Feistel structure was already quite well understood at that time (around 1997, when AES candidates were being designed), it was also quite known to be "suboptimal" in the following sense: to get the most out of the existing security proofs, you had to go to impractical block sizes or number of rounds. Indeed, many researchers were dissatisfied with the Feistel structure, and eager to explore new structures. The AES competition was at the right time to become a test bed for such novel designs, and the accumulated research has shown substitution-permutation networks (as used by Rijndael) to be valid competitors to Feistel structures.