Oral exams and quizzes are hard for reasons unrelated to understanding the subject matter. Language barriers, public speaking anxiety, exam stress, etc. All things that students should hopefully learn how to overcome, but that's a lot to ask a teacher to deal with in addition to teaching history or whatever. With a paper, a student can choose their own working environment, choose a day and time when they are best able to focus, have a constructive discussion with the teacher if they're having trouble midway through the work, and spread their effort (if they want to) across more than an hour-long test or 5-minute oral exam. In an imaginary world where they couldn't cheat, a paper gives the teacher the best chance of evaluating whether a student understands the material.

I don't think you're wrong necessarily, but there are good reasons that teachers like papers other than "we've always used them".