People who consume more artificial sweeteners are more likely to be people who are trying to cut calories, and people trying to cut calories are more likely to be overweight, and people who are overweight are more likely to experience cognitive decline.

Until a study is done which properly accounts for this well known correlation, I will remain incredibly skeptical that artificial sweeteners have any of these supposed negative effects.

I don't know how you define "properly" but the study did account for this, among many other factors:

"[...] We also investigated whether obesity, diabetes, and a healthy diet, all evaluated in 2 categories (no/yes), modified the association between LNCS consumption and cognitive decline by adding a 3-way interaction among the consumption of LNCS, the timescale, and each modifier on model 2."

In the results section: "[...] obesity and diet did not modify the association between LNCS consumption and cognitive decline [...]"

I agree that they should control for this, but I’m not sure if this correlation would hold. I suspect people who are healthy weight are commonly staying there partly by their diet containing low calorie versions of some items. Some things like soda are very hard to enjoy in their non-diet form without drinking a huge amount of calories, so I could see an argument for artificial sweeteners potentially being consumed more in weight conscious people.

Probably correlates with being more sedentary as well. Lack of exercise is disastrous for health.

isn't that easy to fix by simply comparing them to a group who are also overweight and cut calories while still having no artificial suger.