The rod is a fuel saver device. It obviously is adjusted totally wrong.

OK, here is how to proceed. Loosen the setscrew on that rod and slide the barrel along the rod so it doesn’t do anything for now. Then set the throttle pickup point as you are used to doing. EXCEPT the throttle does not start to open when the roller is aligned with the mark on the brass cam. Look to the left of the roller and you will see an upward pointing projection on the intake manifold. Set the spark advance so the mark on the cam aligns with that pointer (not the roller). When the pointer and mark are aligned, that is when the carb should just barely be ready to start opening.

Having done that, now move on to the fuel saver rod. With shift in forward gear, advance throttle to full advance position (bangs up against the projection on top off cylinder block. With spark advance in that position, contact should be made with the set collar on the rod. Further movement of the control will compress the spring on top of the vertical shaft and go "over center". That kicks the carburetor to WOT position. Backing off on the control will partially close the carb while maintaining full spark advance. That is the fuel saver position.

Does the fuel saver linkage work well enough to be considered useful? I’ve always wondered how much fuel per hour it would save.

It depends a lot on the boat it is on. If it has to lug along at WOT to stay up to speed on plane, it won’t help much because it will start to sink down off plane when you back off on the throttle. On the other hand, on a nice boat that planes easily and maintains it’s speed when you back off to the fuel saver position, it does make a difference.