You and I read the same conclusion. You took out the single part that supports your claim and ignored the rest. The whole study is about how socioeconomic status related to crime and that the factors and explanatory models are varied with no consensus among researchers. The idea of the correlation being weak is that you cannot predict criminality risk with only socioeconomic. Instead, the chain of cause and effect is much more inderect and nuanced. Look at this paragraph:

However, a shared feature of all these explanatory models is that socioeconomic background is not in itself viewed as having a direct causal effect on the risk for criminal behaviour, but that this risk is rather affected by a chain of other factors, whose strength varies more or less systematically between individuals located at different positions along the socioeconomic scale.

You are misreading it and possibly misinterpreting what a correlation entails. If I wanted to predict the criminal risk for a person, I cannot rely on socioeconomic alone. That, however, does not mean that socioeconomic reasons don’t contribute to this, and if that was part of your conclusion then I think you are seriously misinterpreting this meta study…