They really do a lot to correct for bias, but is it enough?

I would suspect that a prosperous and thereby complex economy also has more resource to discover or document inscriptions. While they talk about researcher bias, they do not really mitigate this.

And yes, I searched the article and the extensive additional material.

Still a valuable contribution.

The last paragraph is a reference to Cato the Elder. https://en.wikipedia.org/wiki/Carthago_delenda_est

"As a third and final action point, I am of the opinion that Carthage must be destroyed."

So they estimate economic complexity by the number of discovered Latin documents or inscriptions. Wasn't Greek prevalent in the East?

Which is why the title explicitly names the Western Roman Empire, and the article briefly discusses data issues with Greek inscriptions near the end.

can't read it seems

No, based on the professions named within the inscriptions. If a province has relatively many professions that are rare for the WRE, then it's considered more complex. See the paper for the actual estimates, the map in the article does not show the complexity estimation

Wanted to read more on this

Same here - I suppose this article did what it was supposed to do - give a short primer introducing the topic and if you want to see how they computed this look at the paper!

Server died already?

impressive