In practice, RSUs already work that way: you're taxed on RSU delivery, which in the case of a public company is almost always the same as the vest date. Usually the company will sell a portion of the RSUs vested to do tax withholding on the spot, and then you're immediately free to hold or sell the remainder.
I the case of a private company, RSU vesting/delivery means you owe taxes but (usually) can't sell them to pay for it, which is (also) no different from today.
I'm under the belief that vest for tax purposes isn't until the share becomes liquid. If the startup can't exercise then, for tax purposes, has the share vested? I believe it has not (and have, in the past, filed taxes on this belief, with the support of my accountant.)