So the process would be to first do the usual -- joint a face, joint
an edge, rip the opposite edge -- then where you would use a planer,
use the jointer instead on the last face to get the board to final
width. You risk getting the faces out of parallel of course (how
often would this happen?)

A few weeks ago I bought a jointer and tried exactly what you are
suggesting.
I was trying to joint 4 pieces of 6/4 stock down to 1 1/16". It made an
incredible mess of things; some of the pieces were 1/8" out of parallel
before I gave up. Fortunately I still had enough thickness left to salvage
it when I actually bought a planer.

Frankly I don't know if they were out of parallel when they were raw and the
jointer accurately maintained the problem, or if the jointer created the
problem. But, either way it was a fiasco.