Where is the cricket? 6' plus of snow drift, well over flashing,= leaks in winter.
Cap should hang over stone 3/4-2" with a drip edge, missing soft joints around flues = cracked/leaking at every flue.
Demo the cap, and the rubble stone,layout new cricket, flash(100%cover the backup wythe with continuous flashing to the new cap. Replace any damaged flues, flash under new poured and tinned cap or just tin cap the top...to cheap out. Fix the chimney, then install the cricket/new shingles.
Seal EVERY stone anchor at flashing penetration,weeps and drainage screen in cavity.
Don't build chimneys out of rubble. pretty but high maintenance.
Pointing is for walls that haven't fallen apart, this has.