[hervature]

> linear regression on data that is so poorly explained by a linear relationship.

That is exactly what they are saying. This is from TFA:

> The graphics above demonstrate that variation in rates of homelessness cannot be explained by variation in rates of individual factors such as poverty and mental illness.

They are, in my words, saying "Look at this plot, the x-axis has no bearing on the y-axis. To give you a sense of how bad it is, we fit a line to it and it is exactly 0 useful." I don't know why you are focusing so hard on the plots without reading their words. You are in agreement with TFA. Now, for the plot with R^2 of 0.55, that clearly has some positive relationship to it.

As for your last paragraph, I disagree 100%. They are trying to find an explanatory variable, not "summarize the data". By showing all the points, it is evident there is no relationship. As you have continuously pointed this out, the plot achieved its goal. In my opinion, the line is a nice touch for statisticians to know that no illusions from scaling of the axes are playing tricks.