I wonder if it would be a reasonable colonization process to establish a non-breathable atmosphere, allowing postprocessing facilities to later separate oxygen from airborne water vapour.

The notable benefits from such an approach would be the ability to easily deliver asteroid-based water deposits (aim it at Mars, let reentry do the rest), as well as the significant simplification of ground based colonization technology - it's far easier to build resilient habitats for a non-breathable atmosphere than it is for a vacuum, and the risk of accidents and difficulty of venturing outside is much, much lessened (a face mask or filter and oxygen tank instead of a bulky spacesuit), not to mention the radiation protection afforded by a thick atmosphere.