Insulation below fireplace

I assume your gap is on the indoors side?

Regardless, if air is filtering through into the living space, there must be a gap outdoors ( external chimney) as well as indoors.

I recommend you check out the caulking along the chimney to the exterior siding as well as addressing the interior gap. Since this gap inside is directly adjacent to wood trim, I also have to assume that excessive heat is not a factor. Is the metal you are referring to an insert trim of some type? A pic would be a big help here.

If the assumption are right and the gap is 3/4 inch wide, I would use a caulking backer in the gap before using a good grade painter's or acrylic caulk as a topper. The caulking backer is a round foam solid tube that can be cut to length, inserted into the gap to seal it and hold the caulk. After the caulk has cured, it can be painted to match the trim. If for some reason you are concerned about high heat in this area, there is a fire rated caulk made especially for gaps in fire or common walls. Readily available at any box or hardware store from a number of manufactures.

Ok after reading your comments, and learning that the black vertical sides are slate, I would still do similar to above. Perhaps after injecting larger amounts of fire rated caulk as deep into void as possible, then use the solid, tube style caulking backer to fill the majority of the larger, deeper gaps, followed by using some paint-able silicon/acrylic mix caulk.

Before you apply the finish caulk, run a nice straight line of painter's tape up the front of the slate even with the edge of the wood trim. When you fill over the foam backer with the caulk, use a small putty knife to draw the caulk smooth and flat. Immediately remove the tape while caulk is fresh. This should result in a nice crisp straight line. Let the caulk cure for 24 hours. If there is some shrinking or cracks, no problem, simply repeat the process with new tape and another coat of caulk.

Alternately, you could also use a setting type drywall mud, such as durabond 30 or 45 to fill the deeper gaps etc. Then again use a second coat to create a smooth straight extension of the wood trim to the slate.

Either method will result in a neat, easy to paint fill that should stop your air infiltration problem and make the finish look nice.

The advice your home inspector gave you is ridiculous and he should have his license revoked. Either a flue is safe to use with wood, or it is not. Get a good chimney cleaning company that also inspects flues to clean it well first and run a camera down the length looking for any problems that might need attention and assure you it is safe for a wood fire. Don't guess at this or just try a "small" fire before getting a real inspection. Better safe than sorry.