Heading tags (not just the H1) are very helpful from a "what this page is about" perspective and should be thought of as structural elements of a page....for the users! It's not just the H1, use the H2, H3 etc. They help define the content. This CAN help search engines better understand the context of a page.
If you are trying to rank for anything remotely competitive, the H1 is such a tiny tiny signal that it will make no difference. If you are just trying to rank for something really uncompetitive, such as "justin's poinless seo advice" then maybe it will be the little thing that pushes your page above the a competing page.
It's the same as the https factor, it's very weak and only comes into play when the difference in weighting of competing pages is borderline.
Regarding your example....in the distilled experiment they also changed the title tag. Now that is completely different. They didn't test the H1 in isolation. The results cannot in any way come to the conclusion that the H1 had any influence at all. The title tag IS a strong signal which will have completely overshadowed any effect the change in the H1 may have had.
The other experiment....is from 2012!! In 2012 there was no AI, there was no "mobile first", there was no "speed ranking factor" there was no "https factor"....it was a completely different world back then.
Even the Distilled one is from 2016....so it's largely not relevant today. AI has completely changed how we think about SEO today.
I was doing some work on a kitchen website this morning and I just searched for "kitchen manufacturer". Does the top result (Nobilia) for this term have an optimised H1? Nope...it says "Our company".