What's wrong with it is that real programmers constantly use whatever means possible to get the answer to questions as fast as possible. What they describe as "cheating" is what real programmers do all day long.
Unless you are working with a really small API surface then it is unlikely that a programmer is going to have much of the API memorized than the ones he uses on a daily basis. And with people radically changing their stacks every few years it is unreasonable to expect it.
Also, people copy/paste/tweak things from StackOverflow all the time. Disabling clipboard is unreasonable for that reason.
What a programmer can do using only things completely in their head is quite different than what programmers do in reality. Programmers are constantly looking up things they don't know throughout the entire day.
Keep in mind too that online tests are for screening, not a complete interview. You bring people in after that and do several rounds of interviews. Under that circumstance having a friend next to you or supplying you the answers doesn't really get you far if you know you have to do similar tests once you get into the office. There's isn't a lot of incentive to cheat.
The bigger concern is that this is a red flag about the work environment / culture. If I saw that I wouldn't go anywhere near it as well. In fact I deliberately put things in resume that some companies might find objectionable in order to purposely weed out with such cultures.