You and others have noted the numerous deficiencies evident by the photo.
Is this the north/northeast side of house?
How’s the absence of a cricket working?
BS says moisture moves from hot to cold.
Migrating interior moisture may be a part of the cause where the FP breeches the wall/roof. Not sealed as tight as the rest of the house and the path of least resistance.
Also suspect would be the moisture/vapor barrier installs in the CS and at the sill plate and wall sheathing. Are they present without defects?
Kickout diverters at both sides of the chimney would help to some degree along with the other suggestions.
Bottom line is moisture issues have been attacking this structure for a prolonged period of time, thus the growth/stains.