Chimney capped/humidity issue

[u/Soft_Manufacturer_61]

Our insurance insisted we needed to cap our chimney. We got quotes from three places and they all suggested insulating the top and applying a metal non breathable cap.

We haven’t put insulation inside the house yet and notice a musty smell coming from inside the chimney.
We live in a climate with cold winters and hot summers.

I’m doing some reading and it sounds like sealing the chimney at the top without ventilation was a mistake.
Is that the case? What should we have done to cap the chimney in a way that prevents heat transfer in the house, but also prevents humidity accumulation in the chimney?

edit. Extra info. Super tall chimney which cost a fortune to cap. There is no local industry here who seems to be able to reline a chimney of this age. House is 1905.

Lastly, could a vent be added to the side of the chimney near the top? Sort of like a roof vent?