That's correct. Given a polynomial time approximate algorithm for Ek-SAT (note, by approximate algorithm I mean along the lines of the formal definition of approximate algorithm, that the solution given by the algorithm falls in some fraction of the real answer for all instances, see https://en.wikipedia.org/wiki/Hardness_of_approximation), you would show P=NP.

My first concerns, namely the usage of analog methods to solve NP-complete problems, lies along the same lines as this post: https://www.scottaaronson.com/blog/?p=2212

Moreover, from what I can see, and as you mention as your original concern, the extent of proof for the 'exponential-speedup' claim exists in the form of some benchmarks, hardly a proof that their method actually works on all instances, which would be needed to show that it is a approximation algorithm as commonly defined.

The purpose of my post was to highlight that for some problems (for example, TSP), even a really crappy approximation algorithm would imply P=NP. As highlighted by the authors of this paper, MAX-EkSAT is in a similar situation, though judging by https://www.cse.buffalo.edu/~hungngo/classes/2006/594/notes/..., unlike TSP, approximation to SOME ratio is possible, though there exists aratio >1 which cannot be beat unless P=NP.

I was simply trying to address the statement: "So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. ", since an actual approximation algorithm for some ratio really WOULD be significant (showing P=NP).