Just paying the debt per se wont result in score improvement.
However, your score could change depending upon whether the debt was still being reported by the OC, and being used in the calculation of your revolving percent utilization.
Upon payment of the debt, both the balance and any remaining CL still reporting are removed from your % util calculation.
As for your plan to use an attorney for pursuing a good-will deletion, which is what you would do to request deletion of a paid collection, I personally would not take that path.
It makes your humble request for GW appear to be a legal issue. I would keep it straight from the consumer. It is not a legal issue, it is a personal request.