To a certain extent you can dispense with mental logic by using a compiler. But the feedback loop is much slower. Thinking your logic through before feeding it to a compiler is like looking at a map when you're driving a car; you can cut off whole branches of exploration.

Binary search is a particularly tricky logic problem in part because it's so deceptively simple. In a continuous domain it's easy to get right, but the discrete domain introduces three or four boundary cases you can easily get wrong.

But the great-great-grandparent is surely correct that many programming jobs don't require that level of thinking about program logic. Many that do, it's because the codebase is shitty, not because they're working in an inherently mentally challenging domain.