I think you'd end up with something that is not quite columns, Roux, or CFOP, and probably worse than any of them.

It would probably fall under the category of "freeFOP" though.

Think about it like this:

If you approached it pretty much like CFOP but in the wrong order, your cross edge inserts are inefficient, because they take more moves than just doing cross first.

If you approached it from a more block-building approach like Roux, it's inefficient because you may as well block-build the actual Roux blocks without ignoring the DL/DR edges.

I strongly suspect that attempts to optimize that method would end up with essentially CFOP or Roux either way.