Testing if one regressor is a proxy for another
For the benefit of readers I note that, as the authors write at the bottom of p.2557


  "We define trust as the subjective probability individuals attribute to the possibility of being cheated."


So they actually measure "lack of trust" (the higher the value of the variable, the lower the trust), in order to see whether it co-varies positively with risk aversion, and so whether it can be considered a proxy for it.

But the correlation shown between "Trust" and Absolute Risk Aversion in the table you mention is $0.017$ which is not just "weak" -it is essentially non-existent, since with a finite number of (observational) data points, correlation even between random variables that are in reality uncorrelated, will not be exactly equal to zero (even with a sample size of $n=1,444$ as theirs).

This alone suffices for the argument that "Trust" cannot be considered a proxy for Risk Aversion, since in order for this to be a valid argument, we need the correlation between the two to be high.

The fact that Risk Aversion appears "statistically insignificant" means that Risk Aversion has no explanatory power on Diversification, but this cannot be attributed to the presence of Trust, since their correlation is so small.