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).

There is a difference between the ice paddles and a standard bottle -- shape and surface area.

Paddles aren't round, so they'll disturb more of the liquid as you're moving through it, which should lead to more evaporative cooling. They're often either a rectangular or even vaguely X-shaped cross-section, which will have more surface area for a given volume, meaning that it can conduct heat more quickly and cool faster.

Of course, the solution to this is to use more than one round bottle, and then stir with a regular paddle or large spoon.

Oh ... and one recommendation that I've been given for whatever you're using -- bag your container before you put it in the freezer, so you don't have to clean the outside before you put it into your soup. (and based on that, you could probably also put a standard ice pack in a ziplock container and stir with that ... but if it ruptures, you'd likely have to throw the batch away.)

The big difference is sales volume - the mold for the plastic water bottle is moving millions of bottles, the mold for the "Ice paddle" is probably moving 10s or perhaps 100s of thousands of paddles in the same time period. Lower sales volume means higher price per each, nothing magical to it.

Other answers have already covered many good alternate methods - transferring into "hotel pans" on ice is one very effective cooling method. A stainless steel immersion cooler is slightly less effective than a copper one, but also less reactive with a wider variety of foods, so it might be a better choice if going that route; stirring still helps (I also use one for beer wort.) Foodsafe bag(s) or bottle(s) of ice and stirring should work fine, though the bags have some risk of rupture. Immersing in an icebath and stirring also works.

2 or 3 quarts is not what I'd call "large" in this sense - with the beer wort, we're talking 20 quarts so specialized cooling equipment makes sense. Putting your pot with 3 quarts of stock in a larger pot or bowl full of ice, and stirring the stock should do - as would dropping in a smaller metal (for faster/better heat transfer) pot or bowl (clean on the outside) full of ice. Given the "home scale" of that much (or little) stock, rather than seeking out "hotel pans" at a restaurant supply you should be able to fit it in a standard 13x9 baking dish if going for the "spread it out and make it easier to cool" approach. You'd want one with a good tight cover so it wouldn't slosh out - or you can use a "type 5" (polypropylene) flat plastic container (or several as needed for volume) which might be easier to find with a tight seal. The "type 5" plastic takes heat relatively well, so filling with hot stock won't make it warp into uselessness. 2 or 3 smaller pans/containers might be easier to fit in the fridge than one big one, but don't stack them until they are cooled.

Leaching is an issue, even in 'off the shelf food grade packaging' chemicals can leach from the plastic over time, especially when exposed to hot and cold extremes. That's part of the difference in the commercial wands, they're made for exposure to hot and cold temps without leaching. The other mentions as to 'why pro ice wands' are already here, surface area, volume. One other item I do want to address is placing large containers of hot liquids into a cold home fridge. There's not enough air circulation to cool it fast enough, and properly bring down the temperature, but in addition did you realize you're raising the temperature of your entire refrigerator? Other items in your fridge can go out of temp if you're not properly cooling your soup before placing it into a fridge, meats, cheese, eggs and other items are at risk, even those unopened, can go out of temp and start to grow all kinds of bacteria, so just be careful. Not only is your soup at risk of not cooling fast enough, you're putting all your other foods in your fridge at risk as well. In addition, if your soup is thick, and in a large container, even as the outside layers on the side and top cool, they start to insulate from rapid heat transfer, and so the center of your soups will stay warm even longer if not cooled with an ice wand. I would bet that even temping your soup several hours later you'd find it's not down to temp.