So the weird thing is, that people who can’t solve these kinds of fizzbuzz type problems are often employed as programmers.

And, in my experience there are definitely situations where the majority of candidates can’t solve these problems.

Right! I can think of two explanations for this.

One, these non-programmers are somehow not actually doing their jobs, but getting away with it. I've worked with people like that, but only a tiny number. Maybe i've been lucky.

Two, you don't have to be a programmer to be a developer. In this day and age, our tools, frameworks, and resources (ie Stack Overflow) are developed enough that you can actually get some useful things done without having to think mechnically. Knowing some obscure boilerplate (eg Spring annotations), how to use your IDE's autocomplete, and where to go for copy-and-pastable code for various kinds of problem actually makes you a useful team member! Even if you can't write a simple loop to save your life!