It's true that you usually can't cheat your way through such an exam provided you actually write the answer yourself, but in a take-home situation you can always ask someone else to solve it for you.
For things such as proofs, it might give you a problem for which the theorem is needed, then ask you to solve the problem, indicate which theorem you used, and then prove the theorem.
Please tell me how you would structure such an exam without a proctoring system as described.
Of course if you have 50 students per instructor this is not going to work...
Computer science: Solve a complex problem in code. Include a git history. Be ready to defend your program design if I get suspicious.
Econ: Long answer question: Pick 5 concepts that we learned about that you think are most important. Explain them as you would to a ten year old.
I think the larger point is that it is fairly easy in any subject to design a test that is very hard to cheat on. It is much harder to find the resources in modern education to grade that test since each submission is likely unique.
Tests that are easy to grade (like multiple choice), tend to be tests that are easy to cheat....