The same type of reasoning that explains why it makes sense to create a calculator app instead of use one i.e. sometimes it's not the answer that matters rather the skills you learned to come up with the answer that you can now take with you to unsolved problems.

I think it is slightly different. Just rote memorization of the facts can be done, yes. However, these facts require a lot of mechanics of numbers to work. Such that the benefit is less from remembering the facts and more from working them.

Is akin to knowing the idea of how to drive a stick shift, and knowing how to drive a stick shift.

Because breaking out your calculator when you want to figure out whether you can divide 22 students evenly into groups of three is going to get you laughed at.

Divisibility by 13 is of course not as useful/applicable as divisibility by three, but the question is then more nuanced: "Why should I memorize this particular divisibility trick?"

If you know that house number 235 is on the odd side of the road, you've committed a divisibility fact to memory.

This involves less memory than knowing how to use your calculator app to check.

For the exam. After that most of us will quickly forget it.

That way other people can use it too.