Do Ice Wands have an advantage over a homemade solution?
Putting it simply, the nalgene ones are probably only more expensive than regular water bottles because they are more specialized and there is a smaller market for them. That makes it easier for the companies to justify a higher price, especially if they think it will mostly be used for commercial purposes.

For efficacy, you simply want something that will transfer and remove the heat quickly, like a heat sink on computer components. Part of the effectiveness comes from the material used (and its thickness), part of it is the surface area that is exposed, and how much of the cool material (in your nalgene ones, the water capacity) is there to pick up the extra heat before the temperatures have equalized. 

I think the actual difference between a plastic water bottle and a plastic water-filled ice paddle/wand is mostly in the shape (and possibly in the thickness of the walls). Probably, using several frozen food-safe water bottles would be fine, but you might need to agitate them a bit more to get the same effect due to a cylinder having less surface area than the paddles. 

In homebrewing, I use an immersion chiller, a copper coil that gets hooked up to the sink faucet with food-grade tubing. You run cold water through the coil and it spirals through the hot soon-to-be-beer (wort) and copper, being an effective conductor, transfers heat into the cold water which comes out hot at the other end into the sink (or into your garden). It works very well, but it's also pricey because it's copper, and it uses a lot of water (which in the California drought means I need to try to recapture it).

As the comments note, an ice-water bath in the sink also works well. Fill the sink with cold water, add ice, add your pot. Stir a lot in the pot for maximum heat transfer and use a separate spoon or your hand to stir the water in the sink. Replace ice cubes as they melt.

Another option for quick cooling is to transfer it into shallower containers with more surface area and put those in the fridge. More dirty dishes and more chance for spills, but it will let it cool faster in the fridge than in your stock pot (if you have room on your shelves for a bunch of shallow trays).