Listen, your hiring process reflects your company culture. The fact that HR is in charge and you have to work to change it, rather then you, the people who will be teaching and supervising these interns, sitting down, telling HR "This is what we need, This is what we want, how can we make this scale" and having final approval says a lot.

Further, this expectation of a private quiet clean place to interview for several hours seems rather biased against poor people. Not to mention the test itself sounds like it doesn't account for the possibility of people with motor disabilities, the blind, the deaf and other disabled people.

Honestly, open book tests are a thing that has been extensively studied. Use IRT based adaptive multiple choice tests (like the computerized SAT or GRE where tests adapt to your ability), set a time limit, have a rolling set of questions to prevent knowledge transfer, and have an intern test day every quarter. You can calibrate test responses based on Amazon engineers at their desks.