No, the StudioLive Series iii console's mix outputs do not show up as available outputs when viewed in a DAW's I/O setup. The aux/headphone routing is managed as explained in the article above.
The StudioLive mixer does not have software/driver output channels that go directly to the physical outputs (like an ordinary audio interface). When you output from Studio One (or any other software) to the StudioLive output 1 ("StudioLive Ch 1" in Studio One), you are sending output to Channel One on the mixer.