You seem to have a pretty good handle on English. Congrats on that. I think Google is choking on the word "contrivance." To my ear, the word contrivance means "something that was created." So a test is something that was created to see how the student. . . " I think that part of the problem that google is having is that the word "contrived" means "artificial," or "not natural" in some cases. But, I don't think the word is meant to have a negative connotation in this sentence. It just means that Exams are made or designed to see how students solve problems. I hope that helps!

I just want to add that the so-called “translations” and suggestions are inferior to the original, which is accurate English at a high level.

Your first translation, "Exams are the method to see how a student solve new problems." is close. I'd reword it a little. "Exams are a method of seeing how a student solves new problems".