(And yes, the solution is to not exercise until you're able to sell.)
The fear of paying more taxes (or waiting longer for the 1 year period after exercising) causes a significant amount of people to expose themselves to a potentially devastating downside by exercising ASAP.
While this seems like a silly thing to do (and it is, IMO) it has to be seen in the context of a company full of people experiencing (often for the first time) what seems like it will be a very successful "exit" and everyone in the company is in euphoria mode, counting their Lamborghinis before they are hatched.
Of course, after you see the results of this happening the first time you get to be the Debbie Downer who warns people of the possible downside in subsequent companies. IME this is a tough position to be in because you look like a negative asshole when everyone else is in party mode (often being egged on by upper management selling the dream).
Exercise early, pay little tax because the FMV is low.
Exercise late, pay lots of tax because the FMV is high. Exercise late and sell, pay lots of short-term cap gains tax.
Exercising late, while still unable to immediately sell some shares to cover the tax bill you know will come seems like taking a huge risk.