Indirectly, yes. The slight forward rotation of the hips and flattened back causes you to rest on a more anterior portion of the sit bones and you'll scoot back on the saddle slightly in order to position the sit bones on the sweet spot of the saddle. The scoot back is what moves your knee back in relation to the pedal spindle. The slight forward hip rotation also (usually) means you'll probably want to take a bit of the tilt out of the saddle too, but not all of it. The saddle design requires a bit of tilt to function as a hammock.

Reach and saddle set back are really only indirectly associated. KOPS (as a starting point) has nothing to do with reach and everything to do with saddle setback and more specifically, your sit bone position on the saddle. In my case, I can use a zero setback post only on frames with 70-71 degree seat tube angles. For 72 degree seat tubes I go with about 12-15mm setback and with 73+ degree I go with 20-25mm setback. The set back has everything to do with my femur length and desired KOPS position (which for me is slightly behind spindle). As the seat tubes get more steep I need more setback to keep my femur length (knee) oriented to the pedal spindle at the distance I like.

Stem length comes in to play only to the extent that the length of the stem causes my upper body to either work with my saddle position, or against it. When my stem is too short my upper body wants to sit more upright. My back curves and hips roll back (more vertical) and as a result I tend to slide forward off the sweet spot of the saddle and onto the narrow section. The shorter my stem, the more I need to raise the tilt of the saddle to keep from sliding off the front. A longer stem creates a better reach which flattens my back, rolls my hips forward appropriately and scoots my butt back into the saddle. When the stem gets too long I stretch too far forward, roll my hips too far forward (pressure on the perineum) and my butt wants to slide off the back of the saddle.

Given the unusually short reach of the Vaya (considering the reach vs frame size), and the seat tube angle (72.5 for the 57cm), I'm wondering why they would spec a zero setback post. I'm not so sure the Vaya is the best choice for those Jones bars. The bars put your hands pretty much even with the steer tube. Frames deigned for flat bars have longer reach than usual, not shorter. In the end, you might find the only way you can use the SA is with the nose way way up.

-Kedosto