Part of it, as you say, is fat people are more likely to drink diet soda. But this study is even weaker. They tried to remove, via statistics, a bunch of factors (what they ate, exercise, BMI, etc) and still came up with an almost invisible effect:
The difference in heart attacks, etc was 346 per 100,000 person years in higher consumers and 314 per 100,000 person years in non-consumers. A difference of just 32 negative events in 100,000 person years! So a thousand people living 100 years would see a difference of just 32 events...
Every time you apply a statistical filter to remove some factor - lack of exercise, for example - you create error. No one KNOWS the exact effect of lack of exercise, so no filter can remove it with perfect accuracy!
And after all the manipulation for a wide variety of factors, they came up with a difference of just 32 negative events in 100,000 person-years!