Two observations:
1. The test was ridiculously easy. Junior high–level math and reading. And yet teacher candidates still failed it.
2. For K-12 teaching, the knowledge that was tested was pretty much irrelevant for the job. What matters for more than being able to find the length of the hypotenuse of a right triangle or to comprehend an expository paragraph (which is not to say that these aren't useful skills) is the ability to manage a classroom. Almost none of the teacher prep I had was geared towards that single most important skill (which I, unfortunately, was not so great at) and which is what will make the difference between succeeding and failing in a K-12 teaching career.
I think this generalizes to other fields as well. Think about all the tree-traversal BS we've all gone through in a software engineer interview and how little that relates to the actual work that we do and how the hard stuff is much less the programming and more the learning the code base and business.
However, physical therapists already have an occupational licensing exam. You could argue that it’s not hard enough, as it does have a high pass rate for those who have graduated from a PT university program, but then you start drifting into “no true Scotsman” territory.
That's because it's done wrong.
What makes you think that a very difficult exam will improve the ability of your PT?
I think maths is another one, as conventionally taught. I hated maths at school because it was largely sitting down in a room, doing the same thing over and over again like a computer...quite reasonably, I asked myself whether this was a productive use of my time. As an adult, I have taught myself everything again, it was far more enjoyable and useful because I actually took the time to understand the concepts (and no, none of the stuff I did at school was useful, it was just mindless computation).
I don't necessarily think exams are a bad idea but they don't produce knowledge, experience, or real expertise. They are often misused (as is certification, I knew a guy who worked as a fund manager and was a senior member of the CFA for ethics and standards...he stole from employees regularly).
I was somewhat surprised watching the Perry Mason series on HBO last year when Perry Mason was sworn to the bar without actually going to any sort of law school. Apparently that was the norm until after World War II.
It can still be done today in most US states.
Also, note that even the self-represented can take on a government and win. See Jim Pattison's (Canadian billionaire) tax hearings vs. the Crown (Canada), which he won after a decade of litigation. Multiple prosecutors and judges retired during that time period.
Besides developing further tax filing methods for his empire, you can bet he wasn't sued again by the government - they hate taking a loss. (The US SEC infamously tries to undermine defendants today by going after their ability to pay lawyers before trials.)
If by "most" you mean 4 or 5 US States (California, Vermont, Virginia, Washington, and maybe West Virginia). That's 10% of states, although more than 10% of the US population (but still a very small minority).
Meanwhile, anyone can represent themselves in court, and with sufficient facts on their side have won. However, the Supreme Court no longer allows people to represent themselves. This means, in the US there are many cases where you cannot take on the government yourself and win.
Thost states all requires you to spend four years (i.e., even longer than law school) working/studying under an attorney. Washington even charges "tuition" of $2000/year.
I did try to to explain that in my OP:
>>>> (some states have just as lengthy apprenticeship periods replace that)
For example, physical therapy services would suffer if testing were too onerous for patient-interfacing therapists, but the physicians and researchers who set the direction of the PT program can be examined more thoroughly and this is essentially how medical practice is structured.
Similarly, actuaries, whose modeling and underwriting work applies across the entire financial org, can be tested more rigorously than the financial advisors and customer service reps who are busy with customers, custodians, etc.
Raising standards won’t suddenly increase the number of people who can meet those standards. It is likely you will just end up with a lot of shortages for important positions.
The standards have to be considered very carefully to find a balance between ensuring enough quality to be effective but not so high that you can’t fill the need.