I've probably read the same "countless articles" but none of them even referenced any anecdotal evidence, let along a careful study. I agree the study would be difficult because in the time between correcting the NAPs and the search engine picking up the corrections everywhere, there could be other factors that affect your ranking. But if you are doing SEO for 100 local businesses, then you can pretty much determine with some degree of certainty whether it's essentially a myth or not.
So a "published study" to me might be someone like Brighlocal, say... We took 20 local business clients with NAP inconsistencies and monitored their rankings for two weeks on a daily basis and took the 10 of those 20 with the least variation in daily rankings and submitted corrections to their NAPs at the same time using the same method. We performed no other on-page or off-page SEO for the next x weeks. At the end of the "study" the rankings of the 10 businesses changed as follows....
Now of course there's still going to be external factors that can affect the ranking but over time if you looked at similar data for more and more businesses you could be able to present fairly clear data that it makes a difference.
I've been doing Internet Marketing for almost 20 years and over that length of time you gain increased wisdom and gut instinct on things. I'm not saying I can prove the NAP corrections make no significant difference in your ranking, but from all the articles I've read, the claims sure seem a bit fishy. I've seen this before in other industries. The presentation of misinformation, that on the surface makes sense, but really is just designed to make people think they need a product or service. That's the downside of the Internet, anyone can make any claim without basing it in actual fact. Gee, the term "fake news" comes to mind.
Maybe if you could point to one of the countless articles written by someone very well respected in the industry I can pose the question "Specifically how were you able to determine that?"
Actually here's an article I ran across recently. Not sure how I stumbled on it and definitely wouldn't consider the author a widely respected publisher. But he says:
https://www.stlouisdigitalmedia.com/blog/local-seo/what-is-nap-consistency/
"when search engines are crawling the web they can – and will – notice even the slightest variations in a business name, address, phone number, and/or website address. These variations can hinder a search engine’s confidence."
Okay, the first thing that popped into my head when I read this is "what specifically are you referring to regarding _confidence_ and how does that affect my organic ranking if the address and phone are correct but vary slightly from my other listings?"
Or, tell me why this matters so much and how you determined this? Oh, you offer Local Search Marketing Services! Great! I can pay you to help me avoid the issues you are eluding to!
My gut is telling me using general descriptions like "hinder a search engine's confidence" means this author either is basically regurgitating something he heard elsewhere, also with no detailed justification, or is just making stuff up for the purpose of generating business.
Is everybody so sure the Emperor is actually wearing clothes?