Ok, but any techie should have little problem preparing an environment for such a test. Use that freshly formatted, dusted off laptop from college? Maybe even consider using a VM, like others suggested. Or how about install a portable version of Google Chrome, so no existing browser data or extensions would be a cause for concern (or worse, imagine a buggy extension crashing the browser and, thus, ending the test session)?
This is a great point, and I address it at the end of my post. The basic idea is that freedom shouldn't be limited to people who 1) have the technical knowledge to do all this and 2) have the time to set up such an environment. Freedom should be default. By being OK with these kinds of testing platforms, these could eventually be used for non-technical positions very easily, and those applications wouldn't know how to circumvent this.
But Amazon is primarily a tech company. Hate to say it, but non-techies comprise a very small percentage and they're an edge case. I also think that there is little chance that Amazon would care much to impose such strict oversights during non-technical remote tests.
But the real question is: should they have to? Quantumtremor already listed several alternatives that indeed don't have the same problems. Why would you risk ruining the goodwill of your privacy-conscious applicants by making them format a laptop, create a VM, or even install another application. None of that is going to make me want to do your interview.
