I think this comment from djao has some additional insights:
"It would have been nice if TR had asked an academic researcher for comments. After all, if discrete logarithms are ever broken, it'll likely be by academic researchers.
Diffie-Hellman is based on discrete logarithms, but RSA is not. RSA is based on integer factorization, not discrete logarithms. Integer factorization and discrete logarithms share some common techniques, but neither is known to be based on the other, and opinion is split as to whether they are truly equivalent. I can't think of a single respectable academic researcher who thinks that the latest results threaten RSA.
The new discrete logarithm attacks require the existence of a nontrivial intermediate subfield, and work best in small characteristic. They do not apply to the most common instances of Diffie-Hellman in deployment (no subfields, large characteristic), and we currently have no realistic approaches that could make them apply. A similar story actually played out with ECC some 10-15 years ago, when Weil descent attacks were new. Those attacks on ECC also require an intermediate subfield, and 10 years of follow-on research has been unable to sidestep this requirement. To date, there is no indication that Weil descent can be used on the most common instances of ECC being deployed today, and nobody is going around saying ECC is at risk from new algorithms. (Quantum computers, yes, but not new algorithms.)
It is possible that the new discrete logarithm attacks will extend to cases with large characteristic and no intermediate subfields, but I personally think that it is extremely unlikely. I would not put the chances at "small but definite." It would be a major surprise if this happened. Even if it did, RSA may still be safe. I can't speak for others, but my sense of the crypto community is that most experts agree with my assessment."