If you want to suggest a possible causal mechanism, then it's entirely appropriate to suggest that something causes something else with which it appears correlated.

It's not intended to be a proof of anything, just a reasonable and interesting proposal which highlights further lines of investigation.

In fact, the text says " Figure 1A compares the annual incidence of the disease and the household equipment rate in several countries. There is a temporal correlation between these variables."

A temporal correlation is something entirely different to a cross-sectional correlation (arrow of time etc.)

A very simple test is the "Granger Causality", which tests whether one timeline significantly predicts another one. You can formulate more complex models such as time series a predicting b with a certain lag.

Ultimately, the idea is most often to remove unrelated factors (such as control variables, seasonality, self-influence i.e. autoregression) and then measure how well one series at t(0) predicts another one at t(1), while optionally doing some sort of hyperparameter optimization for the lag (i.e. determine which lag works best).

Of course, causation is trickier; roosters don't cause the sun to rise, even if they occur simultaneously (or even slightly before!).