The article doesn't state this well, but the additional cost comes from AMT. If you exercise your options you pay AMT on the face value of a share of common stock at time of exercise minus the exercise price (the spread). This is true regardless of whether or not those options are liquid at the time. For a unicorn, the stock has most certainly increased in value over time, which means the exercise price is a fraction (possibly a very low fraction) of the current value, and you'll be writing the IRS a much larger check than you write the company.
The only escape from this problem is an 83(b) election, which I've heard many companies say they don't allow (IANAL and am not sure what the circumstances are here). It's also the case that with an 83(b) election you are putting real money, potentially a significant amount of real money, into the company's bank account with no expectation of when that investment will become liquid. So this also has risks, but at least the AMT is on the spread, which is $0 in this case.