It tests for multiple things, at its best: A basic work ethic to understand fundamental CS concepts. Sure maybe plenty of people can't write a binary search in two minutes unless they practice live coding a bit, but plenty of people do study, so it self selects for that type.

There are also people who, no matter what, could not live code simple tree traversals or bin search or something, and it filters on that.

Finally, there's a pattern matching aspect to it. Some of the best interview questions I got involved very simple algorithms, but it was obfuscated by the problem. So the 'trick' was to just think through the problem and ask questions. Not to have memorized something obscure.