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I don't think a study like "Aspartame is actually super good for you and makes you run faster" funded by the "American Beverage Association" would ever make it to Theory status, and even concieving of such a silly notion reveals widespread misunderstanding of what a theory is.
On a bit of a tangent, but it's all about positioning, you repeat and broadcast the positive outcomes that you can manufacture supporting data for as much as possible and don't engage with the negative ones. So, we don't even talk about cancer, we just show you how much weight you can lose, and weight loss is obviously good for you, something like this:
https://www.sciencedirect.com/science/article/pii/S0002916523175897
It's real easy to miss, but you'll find a 'Supported in part by the Nutrasweet Company' in the foot notes on one of the pages. The study is not specifically 'Aspartame is good for you and makes you run faster' but... it's pretty close and people are going to draw similar conclusions from it. They don't have to lie, you just have to make sure the 'right' data is prevalent enough that it buries the 'wrong' data.
Well that's fair, but science literate people would then direct you to the largest study ever performed on sweeteners and artificial sweeteners, the NutriNet-Santé population-based cohort study, found the highest risk from any artificial sweetener was Aspartame with a breast cancer hazard ratio of 1.15 which is to say a 15% increase correlation of breast cancer forming in consumers of aspartame over about a decade. The study didn't control for potential selection bias or other outside factors and the CI=95%, range from 1.03-1.22.
Technically speaking, Aspartame would still be preferred for health outcomes over an equivalent consumption of sugar in cases where calorie consumption is high, in the assumption that the Aspartame product in question doesn't contain calories.