I don't have any research or books to link, I did find this webpage
http://chinchillachronicles.com/chinchilla_nutrition_food_and_diet.html that does explain diet pretty well. Chins can't process sugars very well because they aren't designed for a high sugar diet. In the wild sugary foods would be a very rare thing, so they didn't evolve to efficiently digest it (it's generally going to be stored as fat for later use). Too much sugar can cause all kinds of issues including diabetes, weight gain, and tooth decay. Also if too much sugar is consumed all at once the chin can have a dangerous fluctuation in blood sugar which could cause seizures, coma, and death.
I can't find it now (I'm not sure it was online), but I read a study a few years ago about the effects of giving a high sugar diet (in that case daily raisins) to chinchillas. They found that the average lifespan of a chin fell to about 7 years. The primary caused of death were kidney and/or liver failure. I realize you can find people online swearing up and down that it's safe because their chin isn't dead yet or lived until it was 15+, but that doesn't make it safe. It's like looking at smoking or drinking alcohol and saying it's perfectly safe because some people have smoked and/or drank everyday for decades and died a ripe old age.
The problem with those treat sticks, and a lot of other treats sold at pet stores is that they are marketed towards rodents in general. A chin's diet and digestive track is very different then most other rodents. Those ones you linked specifically are bad not just for the honey but also the treat ingredients. Chinchillas also shouldn't be having seeds, they are high in fat. I know chins in the wild do eat seeds, but they also only live 5-10 years and have a very poor quality diet. Where as chins in captivate are fed a high quality diet in the form of a pre blended pellet and have a life expectancy of 15-20 and some live even longer. The chinchilla pellet already contains the a healthy amount/limit of fats, sugars (normally added to make it tasty and as a binder), and other nutrients a chinchilla needs, its a complete diet just lacking enough fiber.
Honestly in moderation a small amount (like the size of a pinky nail) of sugary foods, or fatty foods like seeds and nuts, occasionally is not likely going to do any real harm. However, most people don't understand moderation, and a whole treat stick is not moderately sized. Really most treats are like junk food to chinchillas, they don't really need it, so it's easier to just say none at all. If they never have it they wont miss it. Also just because chins love them does not make them safe or healthy, just like a lot of people love cakes, chips, and candies. There are healthier options for treats, that can be given often and in some cases daily, like the herbs found on
Fuzzies Kingdom's website.