Good question! I think you can come up with different models for how to define a threshold, and I'm not done reading the paper the article is based on. But implicit in any discussion of a threshold is the idea that the threshold has to be explicit. You can't realistically hold a lottery for ostensible fairness maximization without publicly defining your baseline for lottery eligibility.
It would also probably dramatically worsen the grade inflation that's already rampant at a lot of the elite feeder high schools that pump people into the Ivies (and rampant at Harvard itself[1], fwiw).
An institution can make a threshold high enough that a sample task from the admission exam is considered hard and selective enough. It can still have it sufficiently low that the number of applicants passing it is 10-30-100 times higher than the number of admissions, so that the lottery effect is not lost.