Pros: Mavenlink is a PSA with a nice UI and a lot of features that will let you track profitability down to the project level.
Cons: Mavenlink is a separate tool from Salesforce, so the sync is a bit clunky. There's also not very much customizability.
Sync your data to Salesforce and track your team as they move work forward. Set goals for timelines, response/cycle times, work in progress, and more. Any process, any object in Salesforce.
Finding bottlenecks in Salesforce is easy with Flowdometer, a free tool for time tracking and setting SLAs in Salesforce.
Use Flowdometer to be notified of process breakdowns as they happen
If you're not sure where to start with a dysfunctional team, start with repetitive work. Figure out how things are supposed to work, and where they aren't happening this way.