First, it's not necessarily wrong to fit a linear regression to data that might not be from a linear model, or that you know to be nonlinear. The data could be linear enough in the region of interest for the line to nonetheless be useful, for example. Sure, you need an underlying linear process if you want certain theorems and guarantees to apply. But with any data set, linear or not, regression still gives the best linear approximation to the conditional expectation function.
Second, the following paragraph seems to imply that small correlations are the same as no correlation, and the reason the regression is problematic is that the correlation is small:
> How does that fit look to you? I don’t have access to the original dataset, so I can’t check it, but I’m guessing that the correlation there is somewhere around 0.1 or 0.2 – also known as “no correlation”.
But small correlations, if they actually exist, can sometimes be of great practical relevance. So that's not it either.
The actual problem is that the correlation isn't statistically significant – there isn't enough evidence to conclude that the observed (small) correlation actually exists, as opposed to being the result of random noise in the data. And indeed, as some other comments here point out, you can get similar graphs by fitting lines to randomly simulated fake data.
(If you prefer a Bayesian gloss: the data isn't informative enough to move you off any reasonable prior with most of its mass around zero. Same principle.)