In reply to J.C.Ray.

< Does this method apply in case of specification error
The partial leverage regression line indicates whether the LINEAR effect is significant. The curvilinear pattern might be visible in the residual values, but not in the line or its confidence bands.

< I am also wondering about the added value of this graphical tool over the t-value of each regression coefficient
See the previous article for some thoughts on this. If all you care about is the p-value, the graphical method does not provide any advantage.
The general rule for graphical methods is that you expect a graph to provide more insight than a single statistic. The statistic is only valid if the assumptions of the statistical method are met. As you mentioned, diagnostic plots can help diagnose issues such as a misspecified model or heteroscedastic errors, whereas the p-value does not reveal those issues.