This really shouldn't get the fundamental point wrong

> When an employee exercises an option, the company must issue a new share of stock that can be publicly traded.

No. When you exercise, you get the stock, but it's definitely not guaranteed to be publicly traded.

For example Graphcore gave people options, which if exercised became stock in graphcore. If you then found a buyer and asked GC to approve the sale, they declined. Not public. Later they revalued that stock at zero.

To a better approximation, stock options work if you trust the company to pay out.