Instead of spending hundreds of man hours creating a mechanism to better be able to identify what area the problems are in and who to send them to, wouldn't it just be easier to hire more support staff who can channel these problems quicker?

That thinking is short-term. Yes, until an actual program can be written that can identify and channel the complaints to the right source, it would be quicker to hire more support staff. However, in the long term, it's better to automate, even if it's only a partial solution. The investment in programming man-hours to build this sort of mechanism will save the man-hours needed to man the support staff into the indefinite future, especially once you consider vacations, maternity leave, pension plans, retirement funds, etc.

Another added benefit to the automated path is that this application, and the technology used to create it, can be pointed in other directions, to help automate those processes and streamline them, saving us more time and money internally