Well, I'm not a bricklayer, or Mech P Eng, but those bricks don't form the structure of your home. While they provide a façade (or curtain wall) for esthetic purposes, they don't bear roof/building loads. Structural loads are supported by the wall beneath/behind those brinks.
Notwithstanding this, the bricks will experience shear load, that is, the weight loads imposed by their own materials. So, removing a few bricks at a time, and re-mortering in new ones, permitting them to set, then start the process again with a few more bricks would be ideal. No one brick's removal will result in yielding (failing), but mutually (as an assembly) they form a wall sufficient to bear their own weight (shear load).