Why mortar goes first
Mortar is deliberately the weaker material in a wall. It takes the movement, the salt and the freeze and thaw so the brick does not, and it is meant to be replaced periodically over the life of a structure. On a chimney that cycle runs faster than anywhere else on the house, because a stack is exposed on four faces, has no eave over it, and takes wind driven rain that a wall under a roof overhang never sees.
Freeze and thaw, the mechanism behind the damage
The United States National Park Service, writing about common problems with brick masonry, names water as the most destructive force on historic masonry surfaces and states that it wicks into the brick cores, so that in a northern climate where winter freezes are prevalent the brick deteriorates from freeze thaw action. That is chimney decay in this climate. Water gets in during the fall storm season, freezes through the winter and pushes the material apart from inside, so the damage becomes visible in late winter and spring rather than during the rain that caused it.
Matching the mortar matters
New mortar that is harder than the brick around it does not protect the wall, it damages it: the movement and the salts that the old mortar used to absorb get taken by the brick faces instead, and the spalling gets worse after the repair instead of better. Older chimneys were built with softer mortars than a modern bagged mix, so the mix is matched to what is there, along with the joint profile, so a repointed section is not a stripe of gray across a hundred year old stack.
Raked, not skimmed
A repoint that goes wrong usually goes wrong here. Smearing fresh mortar over a failed joint bonds it to loose material, and it comes away in sheets, taking more of the joint with it. The joint is cut back to sound mortar to a depth that gives the new material something to key into, the dust is cleaned out, and the joint is filled and finished to the original profile. It is slower, and it is the difference between a repair that holds and one that fails at the next freeze.
Repointing and waterproofing are a sequence
A water repellent applied over open joints seals the water in instead of out, so the order is repoint first and waterproof afterwards, once the masonry is sound and dry. Doing it the other way round is one of the more expensive mistakes available on a chimney, because the repellent has to weather off before the joints can be repaired properly.
When this work is done
Late winter into spring, after freeze and thaw has opened what the fall filled, and before the next storm season. That is when spalled brick and white staining become visible, and it is the reason masonry work sits in the spring calendar here rather than beside the fall sweeping season.