In Austria, the gyms I went to you just had to sign a form that you know how to climb top-rope, lead, and how to belay.
Most of the rest of the world goes ‘meh, don’t be so obviously dumb then’ and kicks the lawsuits out.
What does that have to do with US gyms requiring belay tests, which is a bunch of steps that doesn't involve "extensive warning in writing".
It lowers insurance bills. If you don't let anyone climb without having done a belay test and putting that paper in a cabinet for 10+ years, then you can get cheaper insurance.
It's the same reason why some gun ranges won't let random people in without joining up and going through a safety intro thing - cheaper insurance.
Which requires due diligence.
Which means there is some guarantee that people belaying at the facility meet some basic standard of skill, so that people are not being dropped all the time and then turning around and suing the facility for negligent supervision/creating a dangerous environment.
In the U.S., for climbing gyms, part of the standard is a belay test. In the article, Mayfield talks about trying to get the industry self-regulating before the government steps in. This is basically how that works: industry founders or leaders establish some procedures for safety, prove them out over time, then insurance companies implicitly adopt them and everyone else follows.
There is a second test for lead but most people take a class and get the lead card during the class.
The coach in the video has some of the worst belay technique I have ever seen. Unfortunately, this is somewhat common among older climbers who learned using the first generation Grigri in the 90s. Petzl's recommended technique back then is very safe (essentially using the Grigri like an ATC), but does not allow giving slack quickly. This made it completely useless for any kind of ambitious sports climbing, and people started coming up with often extremely dangerous workarounds. Petzl has upgraded their recommendations a long time ago, but some people are resistant to change ("it never failed for me"...) Hopefully this video can convince at least some of them to finally adopt the proper technique.