Not really. RSA was subject to attacks because it relies on arithmetic, which has a lot of structure and you can exploit that structure. Thus a number-theoretic advance in factoring becomes an attack on RSA. The reason why ECC is so much stronger on a per-bit basis is because there is a lot less (known) structure to group multiplication on a curve than to integer multiplication. But with the primitives used in symmetric cryptography -- basically substitution or in math terms a permutation -- there is almost no mathematical structure to exploit from general permutations of elements.
Incidentally, one of the concerns that many have about AES is the agressive design and reliance on inversion in the S-boxes which does have some arithmetic structure. E.g. in AES, the permutations are obtained from taking the reciprocal of field elements, so there is some math there. The Soviets famously went in a different direction, generating their s-boxes randomly so there would be no structure whatsoever. Those heavy-duty ciphers are built like tanks and stood up well over time, their only weakness was small block sizes for early versions of the cipher.