According to this:
To summarize: solving the discrete logarithm problem for a composite modulus is exactly as hard as factoring and solving it modulo primes.
So, given your question "Would the ability to efficiently find Discrete Logs have any impact on the security of RSA?" the answer would be yes. Furthermore, if you can solve DLP for composite moduli, you can also solve it for prime moduli. This is, however, slightly different from the answer you linked to since the modulus in DH is not composite. @poncho stated in a comment to that answer that "If your Oracle to solve DLOG works only in a prime modulus, there's no obvious way to use that to factor" which I have not been able to verify.
So, being able to solve DLP for a composite moduli would break both RSA and DH, but it is unclear whether breaking DH would allow you to break RSA.