Hi MrSem,
Yes, the limitations you've pointed out are, unfortunately, baked into the environment. There's no way to prove impact in a vacuum. You can't stop competitors from making changes on their end that could be positive or negatively impacting the rank of your business at the same time that you are cleaning up your citations. Search results are so dynamic, it's rarely possible to be 100% positive in stating causation, with no margin for error, because even if you can control what your company is doing, you can't control what competitors are doing in the same time frame, or as you mention, a Google update occurring in that time frame, or even a bug. So, basically what you have is industry practitioners over time noticing apparent cause and effect. When practitioners state that citation cleanup appears to impact search visibility, it's based on more formal studies like Andrew's (above) and on what they notice across the board with clients, over time.
As for what is actually being cleaned up, that's easier to define. It's going to be resolving inconsistent NAP+W, hours of operation, categories and duplicates, much of the time. It can also include remedying incompletions.
I understand your skepticism. Toothpaste manufacturers have a vested interest in telling you their product will make your teeth whiter. But there's an important difference here that shouldn't be overlooked: the things we've come to think of as local search ranking factors originate in the early study of local search results being performed in the trenches by very small agencies. Going back more than a decade, you can read through the blogs of folks like Mike Blumenthal, sharing what they are discovering moves the needle, client by client. That's not quite the same as the toothpaste company, because the scale was very small at the dawn of Local. You could chalk the corpus of local-related blog posts over the past 12 or so years up to being mere sales pitches, but I think if you start reading backward from 2017, you'll see it really wasn't that way. Just as in organic SEO, it's been a voyage of discovery 