> Why aren’t students tested on their word processing skills?

It depends on the educational system.

> Wouldn’t teachers want their students to be able ... write up a document

It depends on the teachers.

In the early 2000s, I had a similar question nagging me: Wouldn’t schools want all their students to have digital skills?

Fast Forward to 2022, the simplest digital skill (= word processing), is still (as your question prove) an issue (in the US?).