Adding a powered attic exhaust fan will increase the air leakage from the house to the attic and increase your heating costs. References available it you want.
The inspector said "inspection also states moisture in the pit is emanating into the home and into the attic that is causing delaminating".
I assume the pit is either a reference to a sump pit in the basement or perhaps the basement itself. But "emanating" indicated a poor understanding of what could be happening. Winter stack effect moves air from outside into the basement where it is then pushed up through the house and out through upper leakage area, some of which go directly to the attic. Plumbing paths, chimneys, and electrical penetrations provide for a lot of air flow. Those are the paths where the attic fan will significantly increase the air flow.
All air paths from the basement need to be air sealed, link below. In addition, there are many leakage paths that should be addressed from the attic.
Your description does indicate some possible water leak issues, like staining around the chimney. As for the chimney cap, they are only intended to keep water out of the flue, not off of the chimney itself.