I don't have a lot to offer on the substantive discussion on the GRE as I am not informed on the literature nor the larger discussion but in my experience, the GRE at home is more stringent than the test center. When I was taking it at home, the proctor made the environment very uneasy for me. At one point he randomly took control of my screen without informing me, saying he couldn't see my video while precious test time was running out (I lowkey still blame my less than desirable score on that cuz it kind of threw me off tempo afterwards too). He also kept talking to me saying I should do this and that while I was working on the test and that, in my experience, doesn't happen too often in test centers. One of my friends had his test cancelled after the fact with the proctor citing that my friend went away from his screen mid test. My friend claims he got permission to go to the bathroom that but that didn't matter at the end.
Also, in a more general observation, I think cheating on the GRE is kind of counterintuitive as you simply don't have the time to do so. There might be strategic ways to reduce the time spent googling stuff or checking notes but chances are there won't be enough time at the end.