If I understand correctly, customers rate the company in various aspects of the transaction, and then, customers again give an overall score. This is the real-world structure. Making an assumption that customers are reasonably rational (i.e. consistent in their opinions), it means that somehow, they, in their minds, construct some sort of "weighted average" in order to go from the partial scores to the overall score.
Then you should use the regression approach, which reflects the above situation. Using partial correlation coefficients does not capture how one reasonably believes that the customers thought and acted when scoring the company.
This regression is in the spirit of "hedonic index regression", if we view "overall satisfaction" as the "price" of the "product" named "transacting with company", and the regressors as "features" of the product (that are provided in different levels for each customer, and hence their variability).
If the rankings are consistently coded (say, a higher number means a higher level of satisfaction for the partial scores and for the overall score), then a higher estimated regression coefficient on a partial score will indicate that this aspect of the transaction "bears more heavily" (has a higher marginal effect) on "overall satisfaction", and so indeed, focusing and improving on the areas with the higher regression coefficients, should yield larger benefits in overall satisfaction.
But also, in order to finally decide on the prioritization, one should also look how the various areas compare in average score. Say "customer support" has a higher regression coefficient than "delivery", but also, "customer support" is on average rated already very high by customers, compared to "delivery". Then the efforts to further improve "customer support" may be more costly and difficult, compared to improving "delivery". So while one unit of increase in customer support satisfaction may yield higher overall satisfaction increase compared to one unit increase in "delivery", this one unit increase may be more costly to achieve in customer satisfaction than in delivery, offsetting partially, or fully, the economic gains from the increase in "overall satisfaction".
Of course this last issue is not a statistical question, but I mentioned it so that any prioritization suggestion based on statistical analysis, at least mentions this aspect that must be taken into account for the final decisions.